It's odd that duolingo doesn't accept "My car is close", when several other exercises in this same section do accept "close" as the translation for "cerca" as the last word in the sentence. I agree "nearby" is better grammar in this case, I'm just looking at consistency.
You're right. Either nearby or close by makes more grammatical sense. However, My car is close is quite common in colloquial usage. Perhaps report it the next time?
Because "carro" is more commonly used in all countries (75% of speakers), except 4.
If you listen to this sentence you will hear she stresses the es much stronger than the tá!!!