Do your levels and stat records carry over between different region servers?

The stats levels etc is account related not region related. The only reason to pick your region (say America) over Asia is, servers will be near you and thus everyone will have similar ping and you won't be experiencing any lag.

This was an issue when Competitive made it to the Public Test Region, as that region's servers are in the US, people from other regions had ping issues while testing.

OVERWATCH GLOBAL PLAY

Global Play in Overwatch works differently from our other games. All of your game progress, including unlocked in-game items, remains the same across all regions.

Source