I'm with MacArthurBug.
The avoidance of eye contact and inability to read conversational tone or lack of empathy (leading to sarcasm, impatience, etc.)
sounds like a form of the autism spectrum (such as Aspergers).
It doesn't sound much like medicated schizophrenia, or substance abuse or depression or bipolar, for what it's worth.
But really, the biggest point is that there's no way for you to know what's going on for her in her daily life. She may not want the intrusion. (I'm a very different case, but I know when I'm not doing well, I'd rather no one on the outside try and save me. It's well-meaning but misguided.) She may be fine with the relationships she has -- people with Aspergers or autism can have a very low
threshold for social interaction, for instance. She may act like that in church because it's overwhelming -- way too many people. Someone trying to make a connection during what may be the hardest hour(s) of the week could be just too much for her.
This is not to be discouraging to your kindness -- this is just a cautionary note. It
is wonderful that you care for those who are ignored by others.
Still, one of the hard parts of having any mental illness is the misunderstanding of the outside world -- and that can mean they pry too much
or avoid too much. It's also easy to misperceive concern as pity, and pity is unwanted.
If you can talk to her as an equal, and understand that her struggles and desires are not necessarily something you can or ought to help with, you
might make a connection. Maybe you have a common interest -- factual topics of interest will be a better source of conversation than small talk, and if she does have Aspergers or a related issue, a lot of minor social patter/rituals can be difficult and annoying.
You might want to read up a little on adult Aspergers or autism spectrum (try to find a well-respected book or author's website/articles, there's a lot of hooey on the Internet, I'm told by Aspies) if you're curious. Though it's impossible to be sure about what she may or may not have, it might ring true and help you to understand her responses whether or not you make a connection. Don't take it personally if you find she's not receptive at all. Every understanding outsider is still an asset to those who struggle.