"We have a dog."
Translation:Bizim bir köpeğimiz var.
No. We have to use the possessive pronoun here. So we can't say "biz", we must say "bizim".
And the sentence "Bizim bir köpek var." wouldn't normally be preferred, the actual use is "Bizim bir köpeğimiz var." The first sentence is something like informal. You could use it in an informal case, like this:
"Bizim bir köpek var, göreceksin. Canavar gibi canavar."
(We have a dog, and you should see him. He's monstrous.)