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

HD 3 is being developed and tested on Win 7, and it works perfectly on it. It writes the HD_CRASH_LOG.TXT file to the game folder and does not require running it as an administrator from a regular user account, let alone an administrator account. There is absolutely no need to set compatibility with other versions of Windows; moreover, this can negatively affect the online game, up to the point of disconnection.
Other HD users can confirm all of this.

It seems that your HD files have limited permissions. Open the 'Security' tab in the properties of the HD executable files, make sure that your account (user) or user group has all the necessary permissions, and correct them if necessary. If everything is fine with this, then HD is being blocked by some antivirus or something completely unknown is happening.