Verbs of Motion - prefixes on verbs

The Russian Prepositions (Accusative Case) - prepositions of the sort you're talking about

The Russian Prepositions (Genitive Case)

In most cases yes, the idea is на for surfaces, в for enclosures

The second and third links have little tables that lay the structure out.

Looks like this roughly:
towards, located at, from
в acc, в prep, из ------------ enclosure
на acc, на prep, с ----------- surface
к, у, от ----------------------- pillar (often a person)

With кухня, it's a hard thing to come up with a good logical explanation for, especially since the etymology doesn't jump out with an explanation... It's probably best to just consider it an "exception", or better yet an "anomaly"... "That's just the way it is" sort of thing.