Technical issues with Heroes III.

Hello, could you please tell me how to play downloaded maps? I put them in the "maps" folder, but the game doesn't recognize them.

Could you provide a bit more detail: what version of "Heroes" do you have? The game needs a version that "understands" the added maps (i.e., the same or a later version).
It's possible that the problem is that the file name and the map name are different. The game shows the map names in the list when you start the game. Try opening the downloaded maps in the editor - what happens? If they open, select Tools - Map Specifications - Map Name in the menu. For example, the file Valleys of War.h3m is a map with the name "Valleys of War".
And did you, by chance, put zip archives in the MAPS folder? Then, of course, the game won't see them. You need to unpack the archives first (using an archiver or Total Commander).