The preposition "on" is used when there is a contextual reference to the menu being a board or a display. The preposition "in" is used when there is a contextual reference to the menu being a book or a pamphlet. Think of the circumstance where you are walking to the table asking the hostess where you can find out about the salads. She responds as she hands you the menu, "it's all right here in the menu."
I have heard both on and in used in the US. It is probably a matter of which region one is from. Most often, saying something is on the menu means that it is included in a list of items contained in the menu. One might equally say it is in the menu with the understanding that one is referring to the bill of fare contained within the menu. If someone spilled their lettuce and it fell onto the menu, one might say the salad is on the menu. If one then closes the menu on the fallen lettuce, one might say the salad is in the menu. In other words, without a full context both are correct in everyday spoken English in the US.
My point is, in English it means the same; it's a legitimate translation. If you translate the sentence literally, it only says "salad is in the menu", but "on the" is often accepted as a translation of "nel", because that's what it means in English. "off the" is exactly the same in this situation.
No, it is not exactly the same, because "off the menu" is only acceptable with the verb "order". If you are asking whether something is listed "on the menu" is used with the glossy plastic coated menus and "in the menu" with the booklet type menus. Of course, people sometimes prefer one or the other and use them exclusively.
"nel" means "in the" and is used when the definite article for the noun (that which it is in) is "il". For example: nel secchio (in the bucket) "nello" also means "in the" but is used when the definite article is "lo". For example: nello zucchero (in the sugar) "al" means "to the". when the definite article is "il".