I play HoMM 3 using HD mod using 32-bit GDI. As for original HoMM 3, after launching the game for the first time, it is supposed to run 3DO and NWC intro videos and after that run an intro about lore of the game with Catherine Ironfist. However, for some reason it doesn't seem to do so.

I have tried all three options
<Show Intro> = 1
<Bink Video> = 1
<First Time> = 1

Neither of them seems to affect this. Any ideas?

I have the same issue. Wondering the same.

I also tried to add "Show Intro" = 1 and "First Time" = 1 to all HoMM3 registry entries without success:

Computer\HKEY_CLASSES_ROOT\VirtualStore\MACHINE\SOFTWARE\WOW6432Node\Microsoft\DirectPlay \Applications\Hero es of Might and Magic III
Computer\HKEY_LOCAL_MACHINE\SOFTWARE\WOW6432Node\New World Computing\Heroes of Might and Magic® III

Anyone got any idea?

Virtual Store should copy path of regular HKLM path.
Mm... But - without WOW64 section, is it? Im not remember this one.

This is for regular clear original Heroes 3 Complete.

There is easier way to bylass VirtualStore. Enable UAC back to default state (in case if you did change its settings) and run game/hd-mod launcher exe Run as Admin.
This is optional if both GOG Galaxy and game installed onto non-C X/Games/Game-name-123/ location, but mandatory otherwise.