Flying into the Turkish controlled airport you are breaking the law and may have issues with entering the South, especially since you are not traveling on an EU passport.
Paphos area is nice and is likely to be a couple of degrees warmer than the rest of the island but there are few things to do there during a short visit in January-February. Nicosia is the largest city with quite a bit of history and easy (and legal) access to the North. Larnaca is a pleasant seaside town that is not 100% tourism dependent, which means it doesn’t shut down completely during the low season.

if I were you, I’d stay in Larnaca and visit Nicosia for a day. There are regular buses leaving from the promenade that will take you to the center of Nicosia. You can spend a couple of hours in the southern part of the city, go to the North for a couple of hours (bring your passport) and be back to Larnaca in time for dinner