What a wonderfully well-thought and articulated response, Armadius. You're sixteen? You need to become a writer.
Anyway, I think Skar correctly identified my thoughts on that point. However, to be perfectly honest, I would rather have someone in a monogamous gay relationship than have them being promiscuously heterosexual. It seems to me that this one sin has been over-emphasized in proportion to some others. But, let me look at your points one by one.
1. You are right, I think, to question this. The simple fact is that the Bible--as wonderful as it is--really can't be used in this issue extensively. Christ never spoke against homosexuality. The scriptures we can quote on the topic come from Paul or Moses. And, you've already pointed out the problem with the Sodom scriptures--they are by no means conclusive in regards to homosexuality. That leaves us with a scripture in Leviticus where man is commanded not to lie with mankind, and the quotes from Paul (who, in similar passages, is known for commanding that women should not be allowed to speak in church.) So, I must say, you certainly win this point, in my opinion.
The only response I can offer gets us into two more points of LDS doctrine. We take a lot of flack from mainstream Christian churches for these points of belief. 1) We believe that the Bible has a lot of flaws that have crept into it through miss-translations and other errors. 2) We believe that if God has a true church on the Earth, he will lead it by way of prophets to whom he speaks in the same way he did to Moses, John the Baptist, Paul, and Isaiah.
The LDS church was founded by Joseph Smith, who claimed to have seen a vision of God and Jesus Christ. (Story is here, in his own words, if you are curious:
http://scriptures.lds.org/en/js_h/1 Since him, the church has been led by a prophet in turn. (There's a hierarchy, kind of like the Catholics except for two things: the LDS church has no paid ministers, so these men are retired people who quit their other jobs when asked to serve in this position, and seniority leads--there is no electing the next leader, and no politics involved.)
We respect the teachings of these leaders above the things taught in the scriptures. We believe they receive direct guidance from God to lead the church. (We put a lot of stock in guidance from God--like I said earlier, we believe that all people are entitled to revelation and guidance from God to lead and guide them.) So, the prophets (there are fifteen, with the eldest being "the" prophet and president of the church) are very important to us. A book written two thousand years ago can offer great things, but modern prophets are necessary to talk about today's concerns.
So, the modern prophets (I quoted Elder Oaks earlier) have confirmed that homosexuality is a sin. It is because of this stance that the church as a whole can take a stand against homosexuality. (Though, hopefully, an understanding stand.) It is not inconceivable that this stand could change. I don't think it will, but there are times when the church position has altered. The biggest example of this was when the church changed positions to allow blacks to be ordained priests. We believe this direction came from God, and that the world and church had changed enough to allow this.
Either way, we have modern confirmation from a man we believe receives direction from God that church members are not to engage in homosexual romances. We don’t have to rely on the Bible—which is wonderful, but sometimes vague (like it is on homosexuality.)
--Edit, link to Joseph Smith's history added. More responses coming.