Well, technically any ship (NX, Daedalus, Kelvin, ec...) from before 2233 is from the prime timeline.
Indeed. I was thinking of a Prime Constitution Class, but yes, the 22nd Century ones should be identical.
If you're following the discussion on the STAR TREK BEYOND thread, you'll see that there is speculation that the crashed ship could indeed be a TOS Connie.
Although the TOS Enterprise was supposedly launched in 2245 (granted that date is non-canon), there's nothing that prohibits the
design from pre-dating Nero's incursion. The edited establishing shot of Marcus' office) footage from STiD shows such a ship (again "non-canon" -- but if the time and energy was taken to
make the scene, is it still really non-canon? After all, I saw it on a screen!...).
In my view, three possibilities exist: (1) it could be a first-generation Constitution that isn't the Enterprise; (2) it could be the original TOS Enterprise that *was* built in 2245, commanded by Robert April, but lost very shortly thereafter, or (3) none of the above.
I think (2) would be a great 50th present -- and I think Pegg and Lin would love the nod.