If this menu (assuming it's a restaurant menu) is one of those menus that looks like a book or folds out, could you say "quyển thực đơn" or "cuổn thực đơn"?
If not, does this mean that classifiers are actually like noun genders in that you have to remember which ones can or are allowed to apply?
Also, are there specific instances where you would use one form of a classifier versus another, like "cái" versus "chiếc"?
cái is the generic classifier used for a bunch of things from cái này (this), cái đó (that) to being used as a precision marker. e.g. cái thằng này! (this very guy!) chiếc is for transport vehicles and some other things like thongs/flip-flops (chiếc dép), shoes (chiếc giày), socks (chiếc tất/vớ) and even chopsticks (chiếc đũa).
You can say "quyển/cuốn thực đơn" but normally you can just "cái thực đơn".
Some others include:
tấm = flat objects, similar to tờ: tấm ảnh/hình (picture/photo/image), tấm giấy (paper), tấm ván (plank/floorboard)...
bộ = album: bộ ảnh/hình (picture album), bộ tem (stamp collection), bộ phim (film series)...
cây = stick-like objects: cây bút/viết (pen/pencil), cây cột (pole, pillar), cây chổi (broom)...