I'm 19 and my credit score is okay as I just started by my cosigner's credit score is amazing. Will they ask me for a down payment for a car?

Most likely they will ask for a down payment to increase your risk of loss should you default. You are the primary borrower and must meet the lender's requirements for the loan as if you didn't have a co-signer. The co-signer is only the back up plan should you default; they are not the primary source of payments.

Financing a new car on a $1000/mo budget is a tight fit; that's why I expect they will ask for a down payment. It will cost you about $300-$500/mo just to park it in the driveway depending upon how much you finance, the terms, and trim package on the car. Your big expenses will be the monthly payments and full coverage insurance. Add to that monthly expense will be operating costs; primarily gas.

You're making $12,000/yr and a base model Mazda3 runs about $18,000. Even with a cosigner, no one is going to sell you a car that's 1.5x your annual income with no money down. You need to find something cheaper, most likely used.

Would suggest both a lower priced car and a down payment.

yes

Yes, you will still need a down payment...figure 5 - 10% and yes, you will not get the best interest rate.