"The crux of the matter, it seems to me, is simply that the Bible has no sexual ethic. There is no biblical sex ethic. The Bible knows only a love ethic, which is constantly being brought to bear on whatever sexual mores are dominant in any given country, or culture, or period."
That is from Walter Wink's booklet on
Homosexuality and the Bible. It was written in the 90s and is a bit dated now, because he was speaking to those who were not open to LGBTQ folks in the early days of 'don't ask don't tell'.
http://www.religion-online.org/showarticle.asp?title=1265
There are many more books and articles on Biblical scholarship that speak in a more contemporary voice, but this one has historic value as it was the booklet that really opened dialog in the Church, and began the new understandings that the Church is moving towards.
In our own country, the United Church of Christ, the Evangelical Lutheran Church in America, the Episcopal Church, and the Presbyterian Church ordain LBGTQ people. (There is another major mainline that I cannot recall at the moment.) The Moravians are moving in that direction. The Disciples of Christ resist forbidding it. Marriage equality is supported by the four church bodies I named - the ELCA and Episcopalians are working on liturgies now, the UCC has forcefully advocated for marriage equality for a long time. The UCC sued North Carolina for passing a law forbidding clergy from performing same gender marriages.
Certainly one can name church bodies that resist understanding the equality of all people, but it seems that is so much harder to name the church bodies that are open. It is the Spirit moving in the Church. Many smaller non mainline church bodies welcome an ordain LGBTQ people who are called; independent catholic and Old Catholic church bodies in the US were established because of the Roman Catholic Church's stance on gender and sexuality issues. The Metropolitan Church was established for the same reasons with a more protestant flavor. In other church bodies, there are congregations and regional jurisdictions that are pressing their national church bodies to get on the right side of this.
Internationally there are many church bodies, especially in Canada, England, Australia, and Europe, which are on the right side of orientation, gender, and sexuality issues. Africa not so much. It will happen.
The various church bodies in the world are not monolithic. To claim that Christian church bodies reject compatibility between Christianity and orientations that are not heterosexual, that would be untrue. Judaism has its own parallels as do other faith traditions.
Obviously almost all of us have been hurt by things in the past. Some have worked very hard to open the doors to understanding, acceptance, equality. Biblical understandings have been questioned and filled with new insights. Theology has been re-stated, inclusivity under the ethic of love has been lifted up far more extensively.
Being Christian and being LGBTQ are core identities and compatible, in answer to the original post.
The question of Francis cannot be answered with absolutist statements. He has struggled with his own evolving understandings in many areas. He took some regrettable positions as archbishop of Argentina; his current position, being global, has expanded his pastoral ministry. He accomplished a lot of conversations within the Roman Catholic community with his statements. He is the leader of a church body that does not understand the calling of ordained ministry as being open to all of the baptized. He may not be at that point his own self. He is managing within his church body the beginnings of the conversations that have resulted in change and inclusivity in other church bodies. He cannot be said to be 'for' or 'against' anything, but he has gotten the conversation broadly extended. The local permanent deacon in the conservative area where I live has spoken of how Francis shocked people with some of his statements and have begun reconsidering their own closed understandings. He is doing perhaps all he can do, and, perhaps as the Spirit allows, he will do more, but in the Roman Catholic Church change comes slowly; it is a global church which has a block of theology that needs be reconsidered, and that is not a quick task.
For those who want now, and I am one of them, there are the Christian bodies that I have named that are there, now. For those who expect everyone in the Church to be on the same page, now, that isn't going happen. The Church is not monolithic in its earthy expressions and just as we see varieties of opinions, understandings, and life experiences here within the gay male community. We are not of one understanding and opinion on everything; nor is the Church. The Church universal is moving in these regards. There are places now for those who seek the sanction and sanctuary of the Church. There will be more.
For those who wish to live in the sins of the Church, I will not argue because that tends to go nowhere. It is akin to accepting or not accepting the sins of your nation and ethnic heritage. At the root of evil is sin and as we are sinful people in a sinful world, evil happens and sometimes we justify it we slap on approving labels of nation, state, ethnicity, a faith tradition. It is all wrong. Accepting our past and current wrongs can give us grounds to point fingers, or, it can call forth a righteous anger, repentance, and commitment to going forward with justice, equality, inclusion, as God has called us (for those who have a consciousness of God; for those who do not have a god-consciousness, there is an integrity and call of humanism, in all its secular forms, to move forward equally for justice, etc., and we ought to be doing so together, accepting each other, as humans in a world where there is much work to be done).
As I said, being a Christian and being LGBTQ, there is one-ness. We confess and teach that we are made in the image of God. God is not a gender or orientation. God is love, and thus, whatever piece of that image we exist in, we are called to love.