I would think that may vary depending on your location but in general, if you are old enough to drive and have a drivers license and a car, you would be old enough to buy car insurance.
many cities have local auctions for their used police cars usually by accepting closed bids via an ad in the local paper, or you can go to www.govdeals.com or www.publicsurplus.com to get them.
If you buy a car from a dealership you must be 19 so you can sign by yourself. Legal age to sign is 19. That might not go for some others like personal sellers or trades and etc.