Behaviour linked to cage?

[Baron]

Will be taking in another hamster, this time a winter white, at the start of next week. His(her? Not really sure what its gender is) old cage is a small, I think the dimensions are around 50 by 30 cm. I put a picture beneath, maybe some people know what this cage is called?

He's a bit quirky, from what I hear.

-Very tiny. But he turned 1 year old yesterday.
-Lightining quick
-Very messy, pee and poos everywhere in cage
This means his current owner doesn't give him wooden things and cleans his house and wheel weekly.
-Loves to dig, bulldozes his bedding to one half of cage.
-does react to voice, also doesn't bite.
-Can't be handled, very hyper and climbs on and over all his toys
He has a medium sizes silent spinner.

Is this behaviour linked to his cage size?

Also needing name suggestions, he's called a very common and boring name, that translates in to "Tiny face" or "Tiny snout"