> Is there a way to run this setup from a battery or would it need a power socket?

Well, I'm sure there is some way. The power draw would be low enough to run off a battery. But realistically it would be much easier to just plug in every time.

> Could you recommend any portable 4k screens?

No, sorry. I have a 16" 2k portable UPerfect screen which, while not the best screen I've ever used, is very decent for the price, since it was only about $120. But if you're willing to spend considerably more I'm sure you can get much better.