OK, you ask them for a copy of their certificate of insurance, naming you as certificate holder. They should have NO PROBLEM doing this. But there's no way you can verify that they are 'certified' (by whom? for what?).
And that bonded bit, probably means they have a license bond for the state. It probably doesn't mean whatever you think it means. If they aren't bonded to YOU, ie, you are not the obligee on the bond, then the bonding doesn't mean anything, anyway - because YOU can't call it in.