As @littleadv said in the comments, "In California [and possibly other jurisdictions], buying a used car from a dealership comes with a mandatory cooling off period that cannot be waived." If you are lucky, you live in an area which has such a law, and may be able to use it to demand the dealer take the car back. You may need to be able to quote the specific law and threaten them with lawsuit to get them to comply. Consumer protection groups may be able to help you with this process. Move fast; the cooling off period is generally fairly short.
But "As is, no warranty" means no warranty, as is, buyer beware, you really should get it professionally checked out before paying. If that isn't acceptable, do not agree to those terms; if they insist, shop elsewhere.
You should be able to get a correct bill of sale -- again a local consumer protection group may be able to help you get the seller to comply. If not, saving the credit card receipt (and complaining to the local better business bureau or otherwise giving them a bad review to warn future customers) may be the best you can do.
ADVICE FOR NEXT TIME:
When considering a used car, even with a warranty or from friends, ALWAYS get a trusted mechanic (not the seller's!) to check it out and tell you what it's likely to need before you sign. The standard request is "please go over it as if your kid was considering buying it -- would you let them, how much work is it likely to need immediately, and do you think this is a fair price." That inspection will probably cost you a few hundred dollars. But it may save you from a mistake costing much more than that.
If the seller won't let you borrow it long enough to do that, that means they know it isn't worth buying. Again, look elsewhere.
Never let yourself fall in love with a particular car. There are others. If there's anything odd about the transaction, walk away.
AFTERTHOUGHT:
If the dealer insists the receipt is correct, tell the credit card company you were overcharged, present the receipt as evidence, have them correct or reverse the charge pending investigation, and let the dealer argue with the credit card company. Or tell the dealer that, unless you get a correct receipt, this is what you are going to do; I suspect they'll cave pretty quickly.
