For example: if I start as an Alchemist and change to another class, after I surpass my old level as an alchemist (or even before), will this character still make potions when we rest?

What about other class skills like the Monk's damage resistance. If a Monk changes to another character class, does he retain that damage resistance?

no, you would not retain those skills. I think you could still mix potions, but you would not make new ones etc