My ex-husband is on unsupervised probation. Is he allowed to possess a firearm, or leave the state without prior consent or notification?

What were the terms of his probation that the judge set for him. If this was a felony of course he can not own a firearm,
But it just depends on the rules for his probation, and they will vary by person, by court and so on, He needs to check with the probatoin department if he was assigned to a department, or to the court clerk if not and find out more info from them

Yes he should be able to do both. In NC even if you are a convicted felon you still have the right to posses, within you place of business or home a gun.
If he is on unsupervised probation it was apparently not a very serious charge. But as Chuck said the court can require special provisions.