Re: My car wouldn't start this morning.
Yes and no.
Cars are not people, and they cannot "want" or "not want" to do anything. So no, technically you should not say it did not "want" to start.
However, in real life, we do sometimes feel like these objects are acting a certain way "on purpose" -- even though we do know that it's not at all true.
I feel that way about about a certain photocopier. It "knew" when I was in a rush and always "chose" to get a paper jam. Of course I know that's nonsense, but I would still sometimes describe it that way.
So yes, in fact, you can say the car didn't want to start, as long as you chose to say it that way thinking of the car in human terms, not just as a bad English substitution.
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.