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

In the latest version 3.17f, the turn time starts immediately, not after clicking the checkmark.

Yes, in HD+, it really does. I will fix it.

Also, if the opponent doesn't have HD+, you can't start the game; you have to exit and disable it. It would be nice to automatically disable HD+ if the opponent doesn't have it.

This was done in earlier versions. I deliberately abandoned this. If you play SoD, play SoD (with or without HD).
If you want to play HD+ (which is incompatible with SoD), agree with your opponent.
No one wants Heroes 5 to connect to Heroes 4 and automatically turn into Heroes 4.
HD doesn't affect the gameplay.
HD+ does, fixing the bugs of the original game. And HD+ also includes everything that cannot be used against regular SoD, such as password-protected saves.

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I will prepare a list of differences from 2.98. One will be a brief, clear one, and the other will be detailed.

Added 15 minutes ago

If there is a random town on the map under the flag of a human player, then starting a new single-player game and then starting it again causes a crash with the error "ResourceManager::GetSprite could not find the sprite resource".

Can you send me such a map? I can't seem to reproduce the bug.
Also, you are launching HD with the wrong shortcut (most likely created by some old HD 3 beta). Create a new shortcut (it should not have < Heroes3.exe.ini > at the end of the command line, but should have < #... >.