I am going to give advice that is slightly differently based on my own experiences.
First, regarding the financing, I have found that the dealers do in fact have access to the best interest rates, but only after negotiating with a better financing offer from a bank.
When I bought my current car, the dealer was offering somewhere around 3.3%, which I knew was way above the current industry standard and I knew I had good credit. So, like I did with my previous car and my wife's car, I went to local and national banks, came back with deals around 2.5 or 2.6%. When I told the dealer, they were able to offer 2.19%.
So it's ok to go with the dealer's financing, just never take them at face value. Whatever they offer you and no matter how much they insist it's the best deal, never believe it! They can do better!
With my first car, I had little credit history, similar to your situation, and interest rates were much higher then, like 6 - 8%. The dealer offered me 10%. I almost walked out the door laughing. I went to my own bank and they offered me 8%, which was still high, but better than 10%. Suddenly, the dealer could do 7.5% with a 0.25% discount if I auto-pay through my checking account.
Down-payment wise, there is nothing wrong with a 35% down payment. When I purchased my current car, I put 50% down. All else being equal, the more cash down, the better off you'll be. The only issue is to weigh that down payment and interest rate against the cost of other debts you may have. If you have a 7% student loan and the car loan is only 3%, you're better off paying the minimum on the car and using your cash to pay down your student loan. Unless your student loan balance is significantly more than the 8k you need to finance (like a 20k or 30k loan).
Also remember that a car is a depreciating asset. I pay off cars as fast as I can. They are terrible debt to have. A home can rise in value, offsetting a mortgage. Your education keeps you employed and employable and will certainly not make you dumber, so that is a win. But a car? You pay $15k for a car that will be worth $14k the next day and $10k a year from now. It's easy to get underwater with a car loan if the down payment is small, interest rate high, and the car loses value quickly.
To make sure I answer your questions:
Do you guys think it's a good idea to put that much down on the car?
If you can afford it and it will not interfere with repayment of much higher interest debts, then yes. A car loan is a major liability, so if you can minimize the debt, you'll be better off.
What interest rate is reasonable based on my credit score?
I am not a banker, loan officer, or dealer, so I cannot answer this with much credibility. But given today's market, 2.5 - 4% seems reasonable.
Do you think I'll get approved?
Probably, but only one way to find out!