Health questions yes, they need to make sure that they won't be doing the patient any harm by getting her pregnant. As for background checks, no, they're not going to do those and if they can afford IVF they can usually afford a kid since those treatments can be upwards of $50,000. They may try to ask questions to discern the mental state of the patient but since they're not psychologists they can't know for sure if they're crazy or not. In the case you're talking about, the Dr violated medical ethics on several areas and is being investigated for it.