I have a "difficult" question, and am looking for a simple answer. (Good luck with that)

Actually, it's sort of a 2 parter...

Was Hawkgirl/woman (of Earth-1) ever a full member of the JLA?

Is the Hawkman in current JSA supposed to be the Hawkman that was originally from Earth-1?

I know there has been so much crap done to the Hawks and I'm still so lost over who is what is whom, etc. I kinda get the whole reincarnation thing, but not really, all I know is currently, all the members of the JLA and JSA pretty much accept them being on the teams, but leave them alone.

Could someone kinda give me a simple yet clear idea of who became what? Thanks.