汽车 is the complete compound form of the word "car". That is also acceptable to include.
According to a native speaker, "汽車" doesn't sound good. "車子" was her preferred term for "car" in this sentence.
I wrote "我要卖我的车", and it was marked wrong. But could this be considered correct?
Yes, 要 can mean want, but more often implies "need" or "will". That is, it is more certain than 想. I think they should accept 要 but then show you the version with 想。
Who'd wanna sell THEIR car? Because "(賣)" means to sell, and "(車)" means vehicle. :0