I think they're really two separate uses. In "Zij staat op", the separable verb "opstaan" (to rise, to get up, to stand up) is being used. In "Het staat op het menu", the verb is "staan" (to stand), and it's being used with the preposition "op". (This combination of staan + op would be translated to a form of "to be" in English.) You can tell that the verb + preposition combo is what's being used here because if "opstaan" were being used, "het menu" would have to be its object; and in that case, the particle "op" would come at the end of the sentence, like this: "Het staat het menu op". But that's just a theoretical sentence and doesn't actually make any sense!
Not really, it would sound quite wrong. In Dutch, when things are written down they "stand" on the paper (or in the book, or on the wall, etc.). It's a slightly weird way to phrase it, but that's how it is.
"Het staat niet in het boek." - "It is not in the boek."
"Sta jij op de lijst?" - "Are you on the list?"
"Er staat niets op dit bord." - "There is nothing on this sign." or "This sign does not say anything."
I am a native speaker of American English, and "on the menu" sounds natural to me. There could be some regional variation, but even the British English page of the Oxford Dictionaries Online for menu gives the following example: "politics and sport are on the menu tonight".