How to play Heroes of Might and Magic III online via Hamachi?

try playing not through Hamachi, but through pLan, for example.
When I had connection problems, reinstalling Windows and Heroes helped – there were probably a lot of viruses that made everything work incorrectly. :)

There are no antivirus programs or firewalls.

If that doesn't help, try reinstalling Heroes, and as a last resort, install another copy of Windows...