My take on bumping noses is that's how they say "hi". I do that with my cats all the time. The previous msgs are correct: all the behaviors you asked about are pretty normal.
How can you tell if they're playing or fighting? This takes some getting in close, so watch out, but look at the claws: are they in or only part way out (playing), or are they all they way out and really trying to hurt the other?
Also, watch the biting. Are they just biting down partially, not far enough to break skin? Or are they trying to do damage? I know it's hard to tell. It seems like cats have three bites: play bites, warning bites, and fight bites (defense/aggression/hunter-predator). I don't know that for a fact, I'm not an expert, just what I've observed. Maybe someone here can tell me if I'm wrong or not.
Finally, is the action split about 50/50? (as to who's chasing who, who's on top, etc). Then it's definitely play.