Is it okay to mix 12 and 14 gauge wire?

Mixing 12 and 14 gauge wire is acceptable in exactly one direction: on a 15 amp breaker. There, the 12 gauge sections are simply oversized, which the code has no problem with.

On a 20 amp breaker it is a violation. Protection is set by the weakest conductor, and one 14 gauge segment anywhere in the circuit means the whole thing must be protected at 15 amps.

The practical trap is future confusion: an electrician who finds 12 gauge at the panel may assume a 20 amp circuit and upsize the breaker, arming the hidden 14 gauge stretch. Label the panel if you mix.

Pigtailing the two sizes together is mechanically fine; wirenuts are rated for it.

When in doubt, run the circuit through the right wire size checker and let the weakest link set the breaker.