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Your numbers look doable, but they don't leave you a huge margin. Particularly so if your insurance quote is high (or high-ish), and/or if you do not have other savings/investments to buffer yourself with. When I was 21, I'd definitely have gone with your idea; now, I would not. I think a monthly car cost - payment + insurance - of 15% of your gross income is safe, but I'm probably being rather conservative compared to most folks, though my plan leaves enough in your hypothetical budget to be saving 15% of gross for retirement, also.
That said, having a new car, with a warranty, provides some advantages over a cheaper, used car, even if it isn't the financial planner guru's favorite choice. I drove my last car for ten years without payments and between tinkering and repairs, it was costing me at least as much as a new BRZ would cost if I had financed the whole cost, and it was still old as sin, was starting to get structural rust issues, and always needed more work.
I think your idea of throwing a chunk at the down payment is sound - you are making sure you are always on the + side during the life of the loan and reducing your monthly exposure. Your alternative investment options for that money are not great right now, so it isn't like you're missing out on 10% interest or asset growth somewhere else when you throw it at your purchase price.
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