Mostly, sometimes, and yes, respectively. He usually stays by Hawke's side, but will wander off if something catches his interest sufficiently read: if I think it will lead to comedy or be helpful for advancing a plot. He will act on his own without Hawke present in those circumstances, though he's generally not away from her for long--a couple of hours at the absolute max, he doesn't trust her on her own that much. She'd get into trouble without him, naturally. He will stay away longer if she asks it, i.e. to guard a mutual friend, but he won't be happy about it.

I can give some examples if you like. =)

Teo being himself.
Teo using his own initiative in a situation. (old Hawke username, also old style of writing Teo, but you get the drift)
Teo on his own.

Shorter version: Hiccup and Toothless are an excellent comparison. Usually together, but they each have their own priorities. Their priorities mostly coincide, and Toothless/Teo's biggest priority is Hiccup/Hawke, but sometimes a dragon (or dog) has gotta do what a dragon (or dog) has gotta do.