Yep, class 1 is usually reserved for stuff like hardware store stamped steel wingnuts. Class 3 is more for precision stuff - in many cases with class 3 threads any damage like a scratch or nick can result in a thread that won't screw in. The vast majority of fasteners used today will be class 2, and any standard tap you order will automatically be a class 2 unless you special order it otherwise.
A 3A will thread into a 2B, yep. The tolerance is mostly on maximum pitch diameter for the B (internal, or female, thread) and
minimum pitch diameter for the A (external, or male, thread).
So generally, a 3A (male) will be slightly larger in pitch diameter than a 2A. A 3B (female) will be slightly smaller in pitch diameter than a 2B.
Keep in mind that there's still a range within all classes that is deemed acceptable, so you can actually have a class 2 or even a class 1 that fits like a class 3.
Edit: here's a demonstration for a 3/4" thread as an example - so a class 1 mating thread could have a maximum of about .032" slop in the fit, with a minimum of .002". A class 3 about .010" maximum slop and a minimum of almost nothing. Smaller threads have correspondingly smaller tolerances.