Any reason not to use üzlet here? I thought áruház was a department store and üzlet was a smaller store.
No reason, just the preference of the translator. Though "áruház" means something bigger to me than "store" so you're right. "üzlet", "bolt" are both ok for this sentence.
A nuance: would this sentence mean that in the gap between the store and the hospital a car is driving past, or that the car is driving repeatedly from the store to the hospital and back again? In English it's ambiguous.
The first one.
what is the difference between kocsi and autó?
Maybe accepting 'közt'? For me (as a native) "az üzlet és a kórház közt" is also acceptable.