Answer: There are zero down payment lending programs. You may also qualify for lending programs that require a downpayment if you are purchasing a new car with a rebate or you have a trade in, regardless of it's condition.
Answer: There are no money down programs. It is unlikely that your son would qualify by himself. If he applies on his own he would most likely fall into a First Time Buyer program. Most FTB programs require at least 15% of the sale price, not including taxes. If a parent with fair credit was listed as a co-buyer you should qualify for a zero down program.
Answer: The information provided is not enough to determine if you would qualify for a loan. As far a the down payment, differed down payments (30 days) have to be disclosed on the finance contract to comply with the Truth in Lending Act. If it is not it is considered an attempt to deceive the lender. Most lenders do not allow differed down payments. Based on the information you provided I would guess that you would not qualify for a zero down payment program.