- How to play Heroes 3 online.

What should I do? Please advise!

The bug you described during combat has occurred before, but usually, after a restart, everything was fine... (I assume you've already tried this).
Sometimes, the map itself was buggy... you need to abandon this save and start a new game.
If this happens again, have another player be the server (create the game).
P.S. In general, you need to test different options... check the game and Hamachi versions again (version 1.0.1.5 is better).