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

Really? And why?

This is how the developers intentionally designed it. The function that generates a random number for animations in single-player mode uses a standard function and changes the seed (which affects the generated numbers for damage, etc.), while in online mode, it uses the system time as the random number without changing the seed, so it doesn't affect regular random events. Perhaps this is the developers' hint that auto-battles are acceptable in single-player but not in online mode.

Yes, but it only changes if no actions are taken. And because of this, are there problems when there's a disconnect in the final battle, and the battle needs to be replayed?

It is impossible to exactly replay a battle between players in SoD because the seeds obtained are not saved anywhere, and the random numbers will be completely new each time. However, from a game modification perspective, it is quite easy to remove this use of system time, and then, provided that the players repeat all actions exactly, the damage, morale, etc., will be the same as in the first battle.