HoMM3 HD is a mod (add-on) for the game Heroes of Might and Magic 3.

I have a counter-question. You say that when launching Heroes32.exe_HD.exe, it requires a disc. How does it do that? Does a Windows window pop up, or does the game launch and then a Heroes window appears on top of the main menu? If it's the second option, does this happen with a modified resolution or with the standard 800x600?

It's the second option (Heroes window, then the game runs, but only allows multiplayer, without hotseat and the ability to host). I tried different resolutions.

The most interesting thing is that I moved the same archive with the complete package to the C drive (it was previously on the D drive, like the other Heroes versions and the working XP) - and everything started working, it doesn't ask for a disc, the mod works, it's a miracle...

Some nuances have emerged. When launching HD, the remaining mana of heroes is not visible in battle, and the names of some objects are missing (for example, the basilisk pit in the fortress has no name). Is this happening for everyone, or is it another unique bug for me?

P.S. Well, I think I've figured everything out. The mana wasn't showing because of an old Data folder that I inserted after the new extraction.
The installation problems are not due to the discs, but somewhere in the system. I have two, both XP, I mainly use the one on the D drive, HD cannot be installed from it. If I boot the second system (from the C drive), extract the complete package from it, and install HD, then everything works, and then I can launch it from both systems.