"Who has their dog here?"
Translation:Kdo tady má svého psa?
Not strictly impossible, but unlikely. Certainly not the default and neutral form.
If you have people having their dogs in multiple places and you are asking who has it here and not at the other place, you could use it.
But it is not good for just asking who has their dog with them.
In English, should not be the question "Who has his (or her) dog here?
Yes, that is the traditionally correct English usage, and I would GUESS that a translation using "his" or "her" also would be accepted. But using "their" in this way seems to have become The New Normal, and I would also GUESS that, at least in the US, more people than not would use "their" in this kind of general question.