Doko vs dochira?
どこ is more location so "where is...?"
どちら is more direction so "witch way to...?"
is not explained too well
*which way, not witch.
But you are correct. One more thing to note is that どちら is essentially a more polite way to say どこ in this sort of sentence. Both should be acceptable translations of "where" here.
病院 translates literally as "institution (for/of) the ill" where 病 means "ill" and 院 means "institution."
So hard remembering not to omit particles i.e. spoken japanese often would have a comma instead of は :(
Is there a way to render this without の, e.g. having just 病院 as the topic?
"Hospital's entrance" sounds VERY wrong. If a construction like "hospital entrance" is to be avoided, why not use "entrance of the/to the hospital"?
"Hospital's entrance" sounds perfect.
I put びょういんの入口はどこか and it was marked as wrong. Is です reqired here or is it optional ?
Yes, it's required in polite speech, and you're missing the り in 入り口.