battles are not saved.

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PINOCHET911

Are you really thinking I'm completely clueless? After the battle, a battle statistics window appears, and there's a record of the battle, and it's saved; the battle is recorded, but there's no folder with replays, although I can watch this battle, but I can't watch replays of other players.

If you can watch your own battle recordings, then you should be able to watch recordings of other players - look carefully on your computer for these files (battle recordings) - they have the .sav extension and weigh about 300-350 kb in uncompressed form. If you find them, you'll find the folder too. ;)

Added 2 minutes ago

PINOCHET911

Yes, exactly: My Documents/My Games/Heroes of Might and Magic V/Profiles
I think this location was explained on this forum, or am I missing something?

Added 43 seconds ago
or maybe I need to display hidden files and folders?

Of course - be sure to uncheck the box in the "Hidden files" section - most likely, that's why you can't see them...