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

baratorch, what do user rights have to do with it? If version 3.28 works fine, but 3.42 always needs to be run as administrator, does that mean the version needs to be fixed? I can tolerate it. Is constantly running as administrator critical for the computer and the game?

Okay, then I'll ask: why does 3.42 need to be run as administrator? What causes it?
Version 3.42 doesn't add anything compared to 3.28 that would lead to the difference you're describing. I'm 100% sure of that, and it's true for any version compared to any other version of Heroes of Might and Magic.
Heroes of Might and Magic has never touched the location where save files are stored! The code that writes the save files is the original game's code. The autosave code is the only one and has never changed (except for temporarily saving to a separate folder). This has always been the case, in any version of Heroes of Might and Magic on Windows 7.
The changes you're describing from version to version are not changes in Heroes of Might and Magic, but changes in your operating system! I'm 100% sure of that.

File/folder properties in any Windows are opened the same way: right-click -> Properties. In the window that opens, there is a Security tab. In it, you can configure permissions (for reading/writing/modifying/...) for any users/groups of users.
But in general, something strange is happening with your setup. There should be no need to run as administrator or write files to the Virtual Store for Heroes of Might and Magic and Windows out of the box or pre-installed by the computer manufacturer.