Whether or not he has children
Also public record. Also something he should be willing to tell you. Did he think you wouldn’t find out eventually? Plus, you want to be with a man who is proud of his kids!
If he’s clean… (when you get there)
If sex is on the table, you definitely have the right to ask if he’s good. Heck, you have the right to see the records. Yes, even if you’re going to use condoms. Condoms don’t stop everything.
What he does for a living
If he won’t tell you what he does for a living, it’s either because he A) Doesn’t do anything B) Does something strange and/or illegal or C) Doesn’t want you to know he has a lot of money.
Why his last relationship ended
He doesn’t have to get into too much detail, but if he gets angry or irritated at the question then that relationship must have ended in an ugly way—and he may have been responsible. Either that or he’s still into the ex.