Difference between auto and automobil?
Yep. Auto being less formal.
And, curiously, 'auto' is neuter (as in German, cf automobile f in Italian).
Why wouldn't "Where is the car" work?
Because co is what, not where...
I could choose the correct word for car regarding the gender of the question word
"jak*" because auto is neuter, masina is female, automobil is male.
Czech is not Russian, mašina is a machine, not a car. Also, I do not see a place for jak* in this sentence.