You sure every one is logged in over at the live server? You and your friend need to both be logged in at the same time and then you can challenge; well, at least theoretically. Good luck.
can't play live chess with friends

Don't know Jack about the iOS app I'm afraid, but this article should at least give you the lay of the land. Live is in the second part:
https://support.chess.com/customer/portal/articles/1444882
EDIT: Realized I didn't address your "log in" question. Log in makes it sound all complicated when it's really just a mouse click away. Sometimes folks don't realize that chess.com has two "systems" (I'm not good with computers, please excuse any flawed terminology) going at the same time. If you are playing a live game for instance, and you decide to check the forums quickly since your opponent is taking a long think, you're in for a nasty surprise. Leaving the live server is interpreted as abandoning the game. So likewise, just because a friend is logged onto chess.com, unless they are also over on the live server at the same time as you, you won't be able to challenge them. Hope that made sense....
Not sure why I can't play live chess against people in my friends list? It's saying they can't play live chess with me. Where do I turn live chess on if I'm playing on an iPhone 5s??