For the games that steam sells, but you don't have the steam version, can you play with steam edition users and also get invited/informed of their status?

If a game uses steamworks and VAC etc. to take care of it's multiplayer then you have to use steam anyway, so it doesn't matter.
If a game handles it's multiplayer itself, then you'll see everything in game, so it doesn't matter.

But can your friends invite you and vice versa?

As far as I know, no.

http://forums.steampowered.com/forums/showthread.php?t=1319656

Andyman 64 says I can. Does it depend on the game?

If you read anyman614's post, he says you can use the Steam overlay to chat with friends via Steam - that is different then joining their games via Steam, which requires the game to be distributed through Steam / Steamworks.

Basically you can chat with friends via Steam in a non-Steam game (such as a disc copy of Oblivion) but you cannot invite friends to join you - or join their games via Steam - in said retail game's multiplayer unless that game is sold through Steam.