Sure can. They can refuse to exchange a car the second you sign and accept the car (unless there's a serious problem with the car).
There is no 30 day exchange law. Perhaps you are thinking of a 3 day right to cancel, but even that isn't universal; it depends on the state. In some states, dealers are required to give a right to cancel. In other states, the right to return the car in a few days for a refund exists only if the dealer chooses to offer this privilege, sometimes with an additional fee paid for that option.
I'm assuming you read the contract before you purchased the car and would know what was agreed to. Don't ever make assumptions nor rely on something a salesperson might have told you verbally. The contract will state what you are agreeing to. If you expected a 30-day right to exchange, you'd need to have ensured that was in the contract when you signed all the paperwork.
Yes, unless there is something wrong with the car. Lemon law.
certainly unless YOU can prove it is a " lemon "
Yes, read the contract that you signed.
Yes. You usually have 3 business days to change your mind.