"What kind of dog does he have?"
Translation:Millainen koira hänellä on?
It is not incorrect. However I feel like the possession is weaker if the word order is changed. If you ask "millainen koira hänellä on?", I tend to think that she/he is the owner of the dog. But if you ask "millainen koira on hänellä", I get the feeling that she/he is kind of just holding a dog temporarily.
No, it doesn't. "Hänellä" alone means something like "at/on him/her", and (among other thing) it is used in the expression "hänellä on" which translates to "he/she has" in English.
"Does(n't)" is used in English as an auxiliary verb in questions like "Does he eat eggs?" instead of using the main verb directly like "*Eats he eggs?".