JOEL -
"God hath not cast away his people which he foreknew. Wot ye not what the scripture saith of Elias? how he maketh intercession to God against Israel, saying, Lord, they have killed thy prophets, and digged down thine altars; and I am left alone, and they seek my life.
But what saith the answer of God unto him? I have reserved to myself seven thousand men, who have not bowed the knee to the image of Baal." (Romans 11:2-4)

There were always people, even after the apostasy started, who managed to hold on to their belief in Jesus and some of His teachings prevailed. What the LDS Church claims is that after the Apostles died there was no one left with authority to perform the required saving ordinances or to communicate with God to keep His gospel in a pure unaltered form. So there were lots of Christians(a lot more than 7000), but the gospel became distorted, having been contaminated by the philosophies of men.

MICHAEL - Why do Latter-day Saints teach that the "last days" (dispensation) began in 1830 while the New Testament claims it began in Jerusalem on the Day of Pentecost? (see Acts 2:1,4,12-17).

JOEL -
"AND when the day of Pentecost was fully come, they were all with one accord in one place.
And they were all filled with the Holy Ghost, and began to speak with other tongues, as the Spirit gave them utterance.
And they were all amazed, and were in doubt, saying one to another, What meaneth this?
Others mocking said, These men are full of new wine. But Peter, standing up with the eleven, lifted up his voice, and said unto them, Ye men of Judaea, and all ye that dwell at Jerusalem, be this known unto you, and hearken to my words:
For these are not drunken, as ye suppose, seeing it is but the third hour of the day.
But this is that which was spoken by the prophet Joel; And it shall come to pass in the last days, saith God, I will pour out of my Spirit upon all flesh: and your sons and your daughters shall prophesy, and your young men shall see visions, and your old men shall dream dreams: (Acts 2:1,4,12-17)

By the time Luke wrote Acts, Christ had come and gone; so in his opinion at the moment he said this, it was the beginning of the last days.
But then it was Paul that spoke of a future time(dispensation of the fulness of times) when all things that are in heaven and on earth would at last be gathered together:

"That in the dispensation of the fulness of times he might gather together in one call things in Christ, both which are in heaven, and which are on earth; even in him:" (Eph 1:10)

Any time after Christ's death can be called the last days, but more specifically after the 1830 restoration, we are now at that future time Paul spoke of as the "dispensation of the fulness of times", which is also part of the "last days"

MICHAEL - How come Acts 26:16 state that witnessing the resurrected Christ was necessary for the apostolic ministry? If, as some LDS claim, that the absence of apostles from the earth removed their authority (the "foundation' of 'God’s household'), and thus, LDS claim we must have living apostles, does the physical absence of Christ from the earth today mean that the LDS Church does not have a cornerstone?

JOEL -
"But rise, and stand upon thy feet: for I have appeared unto thee for this purpose, to make thee a minister and a witness- both of these things which thou hast seen, and of those things in the which I will appear unto thee;" (Acts 26:16)

All this says is that Christ came personally to Saul(Paul) to call him to be a minister. It doesn't say it is a requirement for every Apostle to see Christ to be an Apostle. Lots of people saw Christ while He was alive, but they were not called Apostles.
Jesus is always our cornerstone whether here on earth or in heaven. He was still the cornerstone after His death while the Apostles were still alive and directing His church. If this wasn't the case, then His church died the moment He died.

MICHAEL - Moreover, if apostles are necessary to keep the Church unified,
(I) why was the Church of Corinth divided if the presence of apostles assured an united church (see 1 Corinthians 1:10-13) and
(ii) why are there at least six LDS sects (not including "fundamentalists"), three of whom have a set of apostles?

JOEL -
(I)In Corinth Paul was only talking to a few church members who were disputing aspects of the doctrines. Paul was there to make sure conditions didn't get worse. Keeping the church together isn't something that magically happens just because Apostles exist. They had to work at it, which is what Paul was doing. After all the Apostles died there was no longer anyone with authority left to do the job.
(II)There are other LDS sects because there will always be a few who don't want to listen or follow the true living Prophet. But the church and true Gospel of Christ has not left the earth, nor will it as long as we have Apostles and Prophets to lead us by revelation from God.

MICHAEL - In reference to 2 Corinthians 12:12, list some "signs, wonders and miracles" done by Latter-day Saint apostles?

JOEL -
"Truly the signs of an apostle were wrought among you in all patience, in signs, and wonders, and mighty deeds." (2 Corinthians 12:12)

The Apostles travel all over the world to preach the gospel to the world. They inspire millions of members to live the gospel and serve others. They have healed the sick many times. They can speak in tounges and interpret other languages. They prophesy of things to come, they dedicate their entire lives in the service of God.
One of the Twelve Apostles of the church came to our city and before our congregation told us he was inspired to prophecy that in the next couple of years our city would see a significant increase in convert baptisms. A year after he said this our city had more than twice the number of baptisms than the year before. Things like this are wonders and mighty deeds.

MICHAEL - Did the church in the first century have a supreme, _earthly_ head? If so, give examples.

JOEL - Jesus is and was the supreme head of the church at all times. In His absence Peter, James, and John and the other Apostles were the main earthly leaders of the Church the same as our First presidency and Apostles are the earthly leaders today.

MICHAEL - In reference to Acts 17: 3-31, Acts 2:41,47 - does that which make one a Christian (I.e. faith in Christ, obedience to the gospel) also make him a member of the body of Christ? (se 1 Corinthians 12:12) - can a member of your physical body live apart from your body? No. Then, according to Paul, one cannot be a Christian without being part of Christ’s body (His Church). Then, isn’t the "one true church" concept false, as many non-LDS obey the gospel and have faith in Christ.

JOEL - Those scriptures were true when Christ's true church was on the earth at His time. Other Christians nowadays may have faith in Christ but they are not obeying the one true gospel as it was in Christ's time and are therefore not part of the true body of Christ. They belong to many different Christian organizations not all believing in the same doctrines. As the Apostle Paul said:
One Lord, one faith, one baptism, (Eph. 4: 5)

MICHAEL - If Christ feeds and cares for the Church, do you believe that Christianity would sicken and die? (I refer to Colossians 1:24, Ephesians 1:22-23, Ephesians 5:29-30).

JOEL - These scriptures were true for the saints who lived at the time Paul said these things. But because of the apostasy Christianity did sicken and die over time. Christ could not feed and care for a church that no longer believed in what He originally taught them.
Jesus and other apostles predicted that there would be a falling away(apostasy). It and the restoration was something that God knew would happen. (Matt. 24:4-13, 2 Thes 2: 2-3, Acts 20:29-30, 1 Tim. 4:1-3, Acts 3:20-21)

MICHAEL - How come LDS do not use the titles "disciples", Children of Christ and so forth when referring to themselves, as LDS believe they are the restored church of Christ and these are the terms the members of the Church were called in the New Testament (e.g. Acts 6: 1,7 Acts 5:14, Galatians 3:26).

JOEL - "A rose by any other name would smell as sweet". We do use these terms. You would have to attend our church for a while and you will hear these terms used. In fact we even have Sunday School lessons on how we can be better disciples of Christ. Our own Book of Mormon says:

"And now, because of the covenant which ye have made ye shall be called the children of Christ, his sons, and his daughters; for behold, this day he hath spiritually begotten you; for ye say that your hearts are changed through faith on his name; therefore, ye are born of him and have become his sons and his daughters. (Mosiah 5:7)

Jesus said to the Nephites:

"Ye are my disciples; and ye are a light unto this people, who are a remnant of the house of Joseph. (3 Ne.15:12)

"And whoso is not willing to lay down his life for my sake is not my disciple." (Doctrine and Covenants 103: 28)

And our church leaders have said:

"The Book of Mormon is his New Covenant memorializing that grand latter-day endeavor. All who receive it and embrace the principles and ordinances it declares will one day see the Savior as he is, and they will be like him. They will be sanctified and redeemed through the grace of his innocent blood. They will be purified even as he is pure. They will be holy and without spot. They will be called the children of Christ." (Apostle Jeffrey R. Holland, "The Messianic Message of the Book of Mormon", p. 339, 1997)

"A disciple of Christ is one who is following and learning to be like Christ-learning to think, to feel, and to act as he does. Becoming a true disciple of the Savior and following his ways are the most demanding learning objectives we can ever strive to achieve. No other discipline compares with his curriculum in either requirements or rewards. Discipleship demands the total transformation of a person by putting off the natural man and becoming a saint through the atonement of Christ the Lord." (Apostle David A. Bednar, "A Disciple Preparation Center" Address given at Brigham Young University-Idaho, 1 Sep. 2004)

"Perhaps we stray from the path which leads to peace and find it necessary to pause, to ponder, and to reflect on the teachings of the Prince of Peace and determine to incorporate them in our thoughts and actions and to live a higher law, walk a more elevated road, and be a better disciple of Christ." (The Path To Peace: President Thomas S. Monson (April 1994)).

"A truly converted follower of Christ--one who merits the designation of disciple or Saint--could not be casual or complacent in his or her approach to service in the Church nor in keeping covenants and commandments." (Conversion and Commitment: Elder W. Mack Lawrence (April 1996)).

Since our Church is called "The Church of Jesus Christ of Latter-day Saints", we do tend to use the word "saints" more often compared to "disciples". The word "saints" is used over 60 times in the New Testament to identify followers of Christ while the word "disciples" is used only about 30 times.

MICHAEL - Where in the New Testament are the following authorized in the first century church -
The Melchizedek Priesthood (to include the Seventy and High Priests)
The Aaronic (Levitical) Priesthood
The Stake presidency
Assistants to the Twelve
President of the Church
The First Presidency
If you could find no New Testament reference for any of the above, can the LDS Church be a "restoration" of the same Church Jesus revealed in the New Testament?

JOEL - When Jesus was on the earth there was no need for a prophet (Church president). When He was not on the earth He used prophets to lead His people as He did in Old Testament times. So because He is not here now, we have a prophet on the earth again through which He directs His people(Amos 3:7). Peter James and John acted as the First Presidency of the early church.
Other extra priesthood authority and positions we have in today's church were authorized by God through revelation to His Prophets, in order to better manage the larger number of church members we have in these latter days and to be able to effectively conduct the massive missionary efforts we undertake.

MICHAEL - According to 1 Peter 1, the word of God, the "seed" of the kingdom is imperishable. (see 1 Peter 1:23-25) - Did the word of God preached to first-century men and women enable them to be "born again"?

JOEL - The word of God does "liveth and abideth forever" as Peter states and I am sure that for a while, while the apostles were still on the earth, men and women were able to be "born again". But unfortunately even though the word of God is imperishable and unmoveable, people can and did move themselves away from the word of God because of their free agency.

MICHAEL - Apostle Orson Pratt, in the "Seer" February, 1853, pages 23-24 said:

"while we speak of only one God, and state that He is eternal, without beginning or end, and that He is in all worlds at the same instant, let it be distinctly remembered, that we have no reference to any particular person or substance, but to truth dwelling in a vast variety of substances." In light of John 16:13 and 17:3, who is speaking the "truth" - apostle Orson Pratt when he said that "God" has "no reference to any particular person" or the New Testament apostle John?

JOEL -
"Howbeit when he, the Spirit of truth, is come, he will guide you into all truth: for he shall not speak of himself; but whatsoever he shall hear, [that] shall he speak: and he will shew you things to come." (John 16:13)

"And this is life eternal, that they might know thee the only true God, and Jesus Christ, whom thou hast sent." (John 17:3)

First of all let me provide another quote from Orson Pratt:

"There are some points of doctrine which I have, unfortunately, thrown out before the people. At the time I expressed those views, I did most sincerely believe that they were in accordance with the word of God. I did most sincerely suppose that I was justifying the truth. But I have since learned from my brethren, that some of the doctrines I had advanced in the "Seer," at Washington were incorrect. Naturally being of a stubborn disposition and having a kind of a self will about me; and moreover supposing really and sincerely that I did understand what true doctrine was in relation to those points, I did not feel to yield to the judgment of my brethren, but believed they were in error. Now, was this right? No, it was not. Why? Because the Priesthood is the highest and only legitimate authority in the Church, in these matters." (Deseret News, Vol 9, No 51; Remarks by Elder Orson Pratt, Tabernacle, Jan. 29, 1860.)

Orson Pratt was well know for expressing opinions that were sometimes not quite in harmony with official church doctrine, but he was at least humble enough to admit it. But in the context of what he is saying in your quote above, it appears he was making a difference between the personage of God and the influence of God's spirit that can be everywhere affecting everything. I don't see how the scriptures from John contradict this.