It's always a good rule of thumb to keep your insurance company in the loop to any changes in your vehicle. For instance, if you want the custom parts covered, you will need to contact the insurance company to add the additional coverages for custom parts and equipment.
I do not know any insurance companies that will issue a policy without a down payment. From the perspective of an insurance company, a driver who does not put down a down payment is too much of a liability.
Since the policy was active from August 31 through September 22, your son will be responsible to pay the premium on the policy. Every company has different rules and guidelines so there may be a fee for canceling the policy before the term is over.
For most major car insurance companies, you are correct. The state does not give any grace period to have the vehicle covered, but insurance companies will transfer coverage for a time frame - usually between 2-4 weeks.