Is "chasing a fly" an idiom?
No, if it were, it would have been translated by an equivalent English idiom and not directly.
Just curious. How accurate are the pronounciations?
Usually they are good, but there are some problems from time to time. This one has some strange stutter in honíš.
Would 'Why are you chasing AFTER that fly' be a wrong answer?
Because it is missing, I will add it.
Also because no-one has reported it, we didn't have a single report for it, not even from you. Use "My answer should have been accepted".
I got confused about tu-te etc. To me they are in Czech to show sth that is close to me. Why then in English it is that or those?
Tu te and such do not mean close to you in Czech. It just is THAT. If they were close it would be TADYTU, TADYTO, TOHLETO, TOHLE, TOTO. There are many of these and I can see how one can get easly confused.
could not 'pursue' work for 'hledat'?
No. "hledat" means to look for / to search for, while "honit" means to chase / to pursue
would "hunting" be an alternative for "chasing" or is there a distinct word for "hunting" ?
Hunting is lovit.
Why does it say tu is "here" when it means that ?
It's also a synonym for 'tady' ('here').