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

- Can I use the old quick battle, but still use all the other improved gameplay features?

- Can I use the replayable quick battle without magic, against any version on the opponent's side? If not, in what cases?

1. Is the old quick battle the one with the flyer bug (hydra in the hive, etc.)? If so, then no. Only in conjunction with the other fixes.
But all the fixes will only work online if they are enabled for all participants in the game.

2. Replayable quick battle only against replayable quick battle.
The type of quick battle (buggy or not) depends on ImprovedGameplay.
But you can use [F] against any player: attack, see the arrangement of forces, or even move something on the battlefield, and then reduce the time by pressing [F]. But you can't change your army composition, because the opponent can't change theirs either.
And also, against SoD, when attacking the AI, the re-prompt works: quick battle or not quick battle. This is to prevent critical consequences from forgetting to turn off quick battle.