Regarding the diffrent styles you can wear your Balmoral, I think I read somewhere that the Glengarry evolved from the Balmoral because some people pulled it up as high as they could and then made a dent in the middle or something like that.

About wearing a headdress indoors might have a cultural aspect aswell as a religious.
I'm in a Studentenverbindung where the rule (goes back to at least 1815, more likely even further back) is:
A cap is a headdress therefor it belongs on the head.
You don't wear the cap:
in church (unless your with the Flag/Banner)
during anthems (same as in church)
if someone of higher rank speaks officialy
while greeting someone of higher rank
while eating
while carrying heavy stuff

I think I've got it all.
The point I want to make is, that diffrent cultures, not only religions, may have diffrent outlook on such questions. I agree that you might take your hat off as a sign of respect and I would do that with most of my headdress but as I said exceptions are possible.