1) I played Y2 connections and didn't lag at all. Those kind of connections caused lag in other fighting games.
2) I believe I saw the lobby option on network mode, but I didn't go in and check for myself. I'm almost certain it was lobby mode though.
3) Honestly not sure on this one.
4) There is survival mode, where you have to keep fighting foes without hp recovery, with one round victories to decide who wins.
5) You can unlock IP points through story mode, survival mode, score attack, versus mode (person or CPU), and online. I got around this much for each:
Story mode: 5800-6500 IP per arcade mode beat
Versus mode and online: 500 IP per match
Survival and score attack was anywhere from 1000-5000 IP; I completely forgot. I keep thinking 3000 IP or so but I'm not entirely sure.
I found playing with my friend and online is the best way to get tons of IP, but if you want your fancy 2500 IP color before going online, just beat your favorite character's arcade mode.