Is "a héten" always implying that it is "this week"? Or can it mean in another context "during the week" (every week)?
What about "ezen a héten"? Would it be too much emphasize or redundant in Hungarian?
How would you ask "What do you do during the week?"
'Mit szoktál csinálni egy héten.' or 'Mit szoktál csinálni egy hét alatt.'
Are you from Felvidék (for the learners: southern part of Slovakia where Hungarian is spoken)? "egy hét alatt" sounds unnatural to me. "hét közben" is another option though.
Yes, I am...and you're right, "hét közben" is a better option. :D
"Mit csinálsz a hétközben?"
"Kozben" means meanwhile, or chronologicaly inbetween.
What is wrong with "what do you do this week"??
This is more of a point about English grammar, but "what do you do this week" sounds incorrect. "What do you do" refers more to what you do regularly, always... restricting it to this week is contradictory.
Ezzen can go after csinalsz to make it flow better right?
Yes: "mit csinász ezen a héten"