Although I haven't tried the multiplayer yet, I think it's separate. What I'm sure of is that the online part does not intervene in your single player game in any way. The single player part is proper classic single player. No one is going to drop into your session to harass you or anything like that.
Btw, I don't even know which version I got, but I think I never used any of the free horses/weapons and stuff. Used the starter horse for most of the time then found a black-and-white spotted horse in a mission that looked kinda cool and changed to that. So no, you absolutely do not need any kind of extra edition for the single player part, but you might enjoy the extra missions it supposedly gives you (if that's in single player, haven't checked yet)