I hope you don't mind. I am a reluctant agnostic - I want to believe in God, really. I am wondering - what do most liberal denominations believe about the Bible? Do they view it as divinely inspired but written by man and therefore containing errors? Do any liberal denominations view it as infallible or, conversely, as strictly allegory? What liberal denominations are most likely to accept lgbt people? Are there any particular liberal denominations that would be more likely to have a contemporary, or at least non-liturgical worship style?
I'd say that a slightly flexible view of Scripture is a big part of the definition of liberal. However most liberals you find here regard Scripture as either inspired or (very similar) as human beings reporting their experience with God. Because humans don't always fully understand what they've seen, it may have some errors. But we still consider Jesus teachings and the teachings of the prophets as authoritative, so the extent that they are correctly reported and interpreted. Paul's status is more ambiguous in many liberal churches.
In a few groups (e.g. Unitarians) you'll find some who consider large parts of the Bible as fictional. But if you're looking at the "mainline", groups like PCUSA, Methodists, or ELCA, that's unusual. However liberal churches often have a variety of members. A PCUSA church recently accepted an atheist as a member (although that caused a lot of negative reaction, and some procedural changes have been made to make it less likely in the future), and most of our churches have some pretty conservative members, and in fact whole conservative congregations.
The attitude towards LGBT depends upon the congregation. In most of these churches there are annual battles at the national meetings, but specific congregations usually have a specific approach. My own congregation (PCUSA) is quite accepting. We had a gay pastor for years (although I'm not sure how widely that was known outside our church). That's common in the Northeast, but there are PCUSA churches in the South that are accepting as well. In denominations like ours where the national policy doesn't permit ordination of gays (although it's not clear whether this is still true for the PCUSA), accepting churches will often label themselves in some way so you can tell.
Actually, conservative denominations have tended to lead in contemporary services. Many of the mainline have traditional services. It's common, but far from universal, to have more than one service of different styles.
[For the PCUSA, the big debate is ordination of homosexuals. The reason it's a big deal for us is that our local leaders are all ordained. So it's not just pastors: in principle gays can't be deacons or elders, although we'd have no problem with them participating in commissions or service work without being formally ordained. However the exact situation in the PCUSA is currently in flux. The national body has said that churches may decide to ordain gays. This could be reversed by the permanent judicial commission, but it would likely be reinstated by the next general assembly. Unlike the Supreme Court, in our system the PJC doesn't necessarily have the last word.]