Cởi quần, áo, v.v. = to take off shirts, pants etc.
Tháo mũ (nón), kính (kiếng), chuỗi (dây chuyền), dây nịt, đồng hồ, găng tay (bao tay) v.v. = to take off hats, glasses, necklaces, belts, watches, gloves etc.
Mở nút = to undo buttons
BTW the bracketed words are the Southern Vietnamese terms for the preceding word.
"Cởi" can apply to anything you wear. "Mang" means "carry", so no.
There are some gray areas. You can say you wear or carry your backpack (technically, you wear your backpack, and your backpack carries your stuff). So in that case, "cởi" works. But for your wallet, you can only carry it, not wear it, and therefore, "cởi" does not work.
It's contextually understood that it's his own coat that is being talked about, hence, "the man takes off (his) coat". The possessive isn't required due to the context, if he had taken off, takes off or is taking off someone else's coat then the additional details are added. (e.g. Người đàn ông cởi áo khoác của chị ấy = The man (who) takes/took off the lady's coat - where the lady is senior to you but by less than a generation)
Furthermore, a native speaker would interpret this sentence as being something that's occurred, i.e. "the man took off his coat". Another interpretation of the sentence could be "the man who takes off his coat".