"Does the boy have a dog or a cat?"
Translation:An bhfuil madra nó cat ag an mbuachaill?
So, I just looked this up in all of my (growing) personal Irish Library resources ... and some online resources (wiktionary) and found that if I translate this to the best of my ability ... in a very literal sense ... it means:
Exists dog or cat (possession, at) the boy?
Look at another sentence for comparison:
Tá madra nó cat ag an mbuachaill The boy has a dog or a cat
To turn this into a question ("Is it the case that the boy has a dog or a cat?"), we take away the Tá and replace it with An bhfuil.