Is zakat obligatory on a sum of money that was paid as a down payment for an apartment, then was refunded with a portion of it remaining in a post-dated check, and when does the hawl (Zakat year) begin for it?

If the sale contract was valid, the money you paid to the seller became his property. When the contract is rescinded and the money returns to you, a new hawl (zakat (definition) year) period begins from the date of your receipt of it. You add to what you received what remained with the seller and for which he gave you a check, because it has become your property. If a full lunar year passes over this money while it is in your possession, and it reaches the nisab (definition) (minimum threshold) (even by combining it with other zakatable assets), and it does not fall below the nisab during the hawl, then zakat becomes obligatory on you, which is a quarter of a tenth (2.5%).