Is dual-wielding possible, given enough practice and proper weapons?

Yes. It has not only been used historically (European duelists using rapier and dagger, Musashi using daito and shoto) but also in modern times, notably with practitioners of Filipino martial arts- check out the section in this vid starting at about 1:50-

https://www.youtube.com/watch?v=CELN-DQI5qc

Obviously using two hands independently requires a higher level of skill than, well, NOT using two hands independently, but it's definitely doable.

Can you block with one hand and then punch with the other? Can you punch alternately with your hands? Weapons are simply an extension of your hand(s). Block/deflect with one weapon and cut with the other or bury your opponent in a barrage. Of course you can dual wield if you train.

Yes.