The grammatically correct sentence of your three examples is "Everything in that shop is on sale."
"That shop sells everything off" does not sound correct, and "That shop sells everything at discount" would actually need to be "That shop sells everything at a discount."
However, there is subtle meaning in these sentences. If you said "Everything in that shop is on sale.", it implies that the sale is temporary. A regular store would periodically put everything on sale, but will return to regular prices when it's over. (Or if a store is going out of business, saying everything in that shop is on sale is reasonable because it is marked unusually low to get rid of it.)
"That shop sells everything at a discount." implies a permanent discount. Meaning a store that tries to purposely mark everything lower than their competitors. For example, Wal-Mart sells everything at a discount.
You can also describe these types of stores as a noun, "a discount store", as in "Dollarama is a discount store."