Does SignalTap II logic analyzer provide time information?

Yes, again assuming you meet timing. If you pipeline them on the way out, then they'll be delayed a few cycles, but easily accountable for.

There's nothing that measures internal delays in real time, at least nothing that does it easy, nor that does a very good job of it. You could run SignalTap at a higher clock rate, but that's not going to be very fine-grained. If you want to know about the internal timing, use TimeQuest, and understand that there is quite a bit of variance(which TimeQuest models with different timing models). What is it you're trying to do?