There's a sightly different meaning in English. "I would like to have a dog" simply means that I think it would be nice to have a dog. The sentence makes perfect sense on its own.
But your version doesn't really work on its own and seems to be answering a hypothetical question:
"If you could have any animal you liked as a pet, what would you choose?"
"I would want to have a dog"
But I don't know if that difference would be made in Polish.
Both fine, but I was trying to find a context where somebody might say "I would want to have a dog". I don't see anything strange about using "have" here, especially as the question was "If you could have ...?" (and I think that's how we'd normally put the queston - plenty of examples on the web)
The thing is that it implies a condition, which "I would like to have a dog" doesn't. OK, another example might be, "If we're going to keep hens, I would want to have a dog, what with all the foxes round here."
"I wish I had a dog" has a slightly different meaning to "I would like to have a dog". The second simply means that you want to have a dog, sometime in the (perhaps near) future.
Whereas "I wish ..." means that you regret that you don't have a dog now. This is close to "If only ...", and I think a better translation would be "Gdybym tylko miał psa".