Can you Burn Edge (pg. 57) if you only have an Edge rating of 1? The rules do not say, and I am not sure whether the special attributes suffer the same penalty normal attributes do when reduced to zero (your PC stands around helplessly).

Just speculating here, but the NPC Prime Runner version of Hand of God requires burning all the NPC's Edge to survive, rather than just one point. That's the only example I can think of where a character would have an Edge of 0 and still function. I know it's not exactly proof positive or an explicit RAW ruling, but Prime NPCs generally follow the same rules as PCs, right?

If it were my game, I'd be comfortable with saying the PC can go on with Edge 0 (and not be able to spend any, of course), maybe with the caveat that they have to raise it to 1 as soon as they have the Karma for it. Clarification form the writers would be great, though. (I don't think it says anything about that in SR4A, either.)

I don’t have the new book but what’s the karma cost of a NEW attribute? A new skill point is proportionally more expensive than the first time you raise it, would attributes follow the same logic?

Not any more it isn't (unless I read it wrong).
New Active is 2 Karma, New Knowledge/Language is 1 Karma.