In this case, a different word order has a slightly different contextual meaning. The further adjective modifies the part closer to the noun. "nové hezké auto" says to me that he already has a nice car, and now he has another nice car, which is new. "Hezké nové auto" says to me that most importantly it's a new car, which also happens to be nice.
In American English I think it's technically correct but not natural, and in my opinion it's fair for Duolingo to reject such answers.
"auto" does mean "car" but is rarely used on its own like that. It's usually used in noun-as-adjective form, like "auto repairs" or "auto insurance." Occasionally also on signage - for example, I can imagine a sign on a toll booth that says "autos $1.00."