What happens if I leave Moldova through Transinstria? I guess I won't get a Moldavan exit stamp, would that be a problem if I visit Moldova again? Wouldn't it look like I overstayed in the country illegally?
Moldova is a former Soviet state, and they take immigration violations seriously. However, when entering Ukraine you'll get a Ukrainian entry stamp, which, in practice, will show you left Moldova.
That said, if you're an EU/Schengen citizen, you can also enter Moldova using a national ID card (except the old Italian ID, which is not accepted), eliminating the need for stamps.
It definitely will (how serious is a distinct question.) Years ago, you could get an entry stamp in Varniʈa, but quick googling suggests last year Moldovian border guards got a permanent representation directly at Ukrainian checkpoint in Kuchurgan (can't find a decent official statement on this subject.)