Usually it's hard to discern if someone has a mental condition, it's not something they like to flaunt around until they are certain they have gained your trust.
Bi-polar generally I wouldn't get involved with, just because of bi-polar people I've known in the past and how illogical and discontented they are ALL THE TIME. Plus they're general tendency to be extremely bitchy.
Depression or major depression can work. They'll actually probably love you and be more faithful because that you take the time with them knowing they've got baggage. But you need to be honest with them. Once they tell you something about themselves and you can't deal with it, don't pretend to be okay with it. You'll upset them and yourself more than needed. Just get out and remain friends. It'll hurt them but they won't get as hurt as they would eventually down the track.
Eating disorders are easier to deal with, because it's almost purely body issues they have, not mental ones. I don't know about getting involved, I guess if you would support them and prove to them weight isn't everything.
There's my two pennies.
And yes, I still consider pennies an adequate Australian currency =p.