All religions have vast differences within them as religions are made up of people and people by nature can be very kind and very mean, not to mention selfish and self righteous.
Tekla, you mention that you have not seen a more diverse group of people. I have no clue where you live or anything else about you, but I can say that the Catholic Church, like many of the extreme religions, are anti trans and only slightly better at gay and lesbian. I have no clue what gay and/or lesbian means to a pre-op trans person, so I will just say trans.
Okay, the Catholic Church is somewhat accepting of transsexuals, as I understand it, if and only if the person who converts does not have sex date or do anything else intimate with anyone else. Even then...
I cannot speak for any other corner of the world, but here in what someone in my family has repeatedly called "The gay capital of the world," namely Los Angeles, which is hardly a gay capital, the extreme religions are just not trans friendly. Chabad is anything but trans and gay friendly. Let me see, words and phrases like "should all get put on an island" and "locked in jail" come to mind. Yes, I did not talk to "every" Chabad person, but they make it a point to say that they are traditional and each is like the other.
I have talked to a couple Born Again Christians and they are nice to talk to, if one can say that, but they are uber, no to mention ultra, anti transsexual.
Yes, I am sure that there are individuals who are more understanding, but by and large many are not.
The Reform Jewish movement seems to be making strides towards LGBT acceptance. There are Christian denominations that also accept LGBT or at least LGB. I am not sure if the Episcopalian Church is trans friendly.
You did not mention JFK, but he was a Catholic and pushed Civil Rights, but I have no clue if he would have been for or against LGBT Civil Rights.
I still find it weird that T is in the same acronym than LGB. I must be different. When I was young and knew that I was a girl, how I wanted to have sex was not in my immediate thoughts. I just wanted to be a girl, a real one. The realization that that means having sex as a girl or boy just did not enter my head. I was way too preoccupied just wanting to be a girl and thinking of girl things. Yes, intercourse comes with the package and the alliance is a natural one now that I see it, not to mention helpful, but still. Yes, off topic, but the time is almost midnight, so why not.