Expressing "I have (got)..." or "Have you (got)...?" is a little contrived. It's almost as though they'd forgotten to include the word "to have" and so came up with a work-around.

To say "I have a dog" it is literally translated as "My-dog it is (exists)"

We therefore use "van" (is) and would translate literally as:
Van kutyám. Is (exists) my-dog.

Generally:

• Van kutyám. I have a dog.
• Nekem van kutyám. *I* have a dog.
• Kutyám van. I have a *dog*.
• Értelmetlen! Nekem van a kutyád. Oi! I have *your* dog (so come and get the damn thing!)

When more than one item is mentioned (qualitativly, not quantitativly), we use vannak instead of van.

Note that if we have an unknown plural number of dogs, we use the multiple objects case; but if we have a known plural number, we use the single objects case.
See plurals page.

• Vannak kutyáim.  I have dogs
• Van három kutyám.  I have three dogs. (Not kutyáim, note!)
• Három kutyám van.  I have *three* dogs. 

Simply turn the statement into a question with a question mark (and a penultimate-syllable rise and then fall of intonation).

• Van kutyád?  Do you have a dog?
• Vannak kutyáid?  Do you have dogs?
• Neked van kutyád?  Do *you* have a dog?
• Három kutyád van?  Do you have *three* dogs?

We form the question "who has?" by using the dative form of who and being explicit with van.

Ki = who

-nek = dative ending, which is key in possession.

• Kinek van egy kutyája? Who has a dog?
• Kinek vannak kutyái? Who has dogs?

Note how dog(s) is placed into the third person possessive case.