Hey! I think you are doing a great thing by coming here and getting started on your budget. The best of luck to you.
As you have time to play with the numbers and see what works for your family, I would just put off the car issue for now. No need to worry about that for a while! Although it is great to plan, there is no need for worry!
I don't think that it is wrong to pay for a teenager's car. We paid for our older son's car, which we bought new. We pay for his insurance and we still give him a fifty dollar a month allowance. Many will not agree with this, sometimes we wonder about our sanity as well!

But, he goes to an out-of-zone school for an academic program ~ no transportation ~ plus, he plays varsity baseball~ in Florida ~ they play ALL year. He also has a job, but in the spring, he can only work Saturday nights. In our county, High School releases at 3:45, which is late, but it leaves little time for work and baseball because of the state rules about how late kids can work. He is VERY FORTUNATE to have a job. (Being willing to work Saturday nights helps! Fast food bosses love that!)
Do we teach him about money? Yes. Today, I talked to him about a bill we had had with 0% interest. We talked about minimum payments, how long it would take to pay the bill etc. if we hadn't paid it off by the due date.
Are we getting through to him? He hasn't touched his allowance in over 6 months. He saved 40% of his earnings last year. I know it is subsidized by us, but he is learning how to stretch his 70 dollar bi-weekly paycheck. He goes nowhere ( movies, mall ) and spends money on nothing except for gas. (2.84 a gallon) We tease him about having a girlfriend, but he says he can't afford it! (How he will pay for prom for him and his date is beyond me! Savings, I'm sure. He hasn't asked us for money.) He also volunteered to pay for part of his summer baseball fee! He wants to work to pay for it and I am so touched by his willingness to do that! (Only three more weeks of spring baseball)
Sorry to ramble, but I just wanted to point out that just because a child is given something, doesn't necessarily mean they don't appreciate it or value it. If you guys plan and find you have the money for the car after you have paid the bills and saved for retirement, then you can do what's best for Your family with Your money.