Thanks so much for all of your help, everyone. They are open today, so I called and asked to speak with the finance guy's boss. He won't be in until Monday.
Another part of this: The finance guy forgot to ask me to sign an arbitration agreement. He fed-exed it to me and said I need to sign it. And I can return it with the receipt for the tire that blew and he will reimburse me.
From what I gather, the dealership should not have been able to let me leave with the car without that being signed. Maybe I have some leverage to have them redo the loan?