How old do you have to be to buy arrowheads?

New or old?

Arrowheads alone constitute no threat. Therefore, I would say you would have to be old enough to reach the top of of the counter to pay for them.

What did the dealer say to do? Did the letter provide any details besides you are declined?
I am going to tell you exactly what NOT to do, do not engage in a bunch of talk with the dealership, I think they knew exactly what they were doing here and have a plan.
They are the experts and have been in this situation many times.
1. Read your contract, word for word. By that I mean every single sentence. What state are you in? Is it a generic contract that they agree to, but allows them to assign the contract to a lender? If so then you say OK I will mail you (the dealer) my payments every month. Right, you don't care who you pay.
2. A car sale can not be reversed, a car is either sold or not. Period. If they start using legal terms or threatening repo, get an attorney immediately.
3. Just get an attorney anyway. Let them handle this.
4. If they want the car back you say OK, but they eat the $4,000 negative equity, keep in mind they had to pay off your loan to get the car title. And something really odd here. Most lenders can not turn around a title in 14 days? Maybe if an electronic title? Meaning if I pay off a loan today it will take the lender 3-6 weeks to process the payoff and get the title out.
Go to the DMV, tell them you sold a car and want to see if the other party has transferred the title yet.
5. Call your lender on the trade in. Ask if the loan has been paid off. DO NOT tell them about this problem, you are just checking since a large check was mailed to them, no worries.
6. If they have paid off the loan and want the car back, then they are eating the -4k. If they offer that then take it, you just have to get a cheaper used car.
7. I bet they will say we have another car for you, a much cheaper car that will be way over priced. DO NOT sign any new contracts.
Best of luck.

You would pay cash for a $1400 car. You won't get a loan for a small amount like that. chances are the car will be in the junkyard before a loan would be paid off. $1400 doesn't buy much.