In the Old Testament there were many examples of proselyting prophets and individuals. That was ALWAYS a big part of the bible.
In the New Testament Christ tells His followers to go into all the world teaching repentance and baptism. Outside of the gospels and the revelation, nearly every dealing in the NT is in reference to this great missionary effort.
It is therefore absurd to assume that the bible instructs us not to proselyte or share our feelings; in actuality it not only shows us a pattern of that, but commands us to do so.
Beyond that, inside of the LDS church we do have temples, we also have a university system. To get admitted to BYU you need to have good grades to justify you being there and remain in good standing with the church. One cannot come in to a class without being admitted to be there. On a much more important scale the temple also requires certain covenants and knowledge to have been gained by an individual as prerequisites to it's attendance.
Nobody is barred from entering the temple, the goal of LDS missionary work is to get people and families to the temple - thus rather than people being kept out, we have 60,000 full-time missionaries trying to get people to be able to go to the temple. But there are steps between that starting point and the temple that need to be taken, and there are tons of people helping along the way. Unlike a university that can only take so many applicants, EVERYBODY can and is encouraged to be prepared and worthy to enter into the temple.
Part of that preparation and worthiness for the temple includes chastity and virtue. In the temple we make certain promises and covenants, among them we make a covenant of chastity and fidelity - thus it would make little sense and be effectually mocking God for an unchaste person to be continuing his covenant of chastity. Most sins are hard to stop once started, outside of addictions, I can't think of anything as hard to stop as sexual activity would be once started; so we take that very seriously.
Back to your question on forced conversion, there is a teaching and interviewing process to ensure that conversions aren't forced; there are dozens of verses in the bible about going out and sharing your beliefs, even more in the Book of Mormon and other scripture. But logic as well would let you know that if you had a wonderful knowledge that brought peace and happiness through the joy of Christ, and this filled you with a love of others, then it would be cruel not to go out and share what you know and what you have.
So it's a love of God and fellow-man that motivates us to share our beliefs with others in addition to the commandment in scripture to do so.