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

1. How will RC7+ImprovedGameplay interact with 2.98? Which version will have its bugs fixed?

2. Am I correct in understanding that in the RC7+ImprovedGameplay+TournamentExtensions vs. 3.0 scenario, if the other player is using an earlier version of RC7, the game will run without ImprovedGameplay and TournamentExtensions?

3. Is it possible to implement (if it hasn't been already) a notification system in the game between the current versions of the mod, informing both players about the status (on/off) of ImprovedGameplay and TournamentExtensions for their opponent?

1. RC7+ImprovedGameplay vs. 2.98
From the RC7 side, no bugs will be fixed, as RC7 will detect the other player using an older version (since 2.98 doesn't identify itself on the network) and will behave as a standard client.
From the 2.98 side, it will depend on its "Fix gameplay Bugs" option, as 2.98 doesn't have any way of knowing what's on the other end.

2. Yes, but only for those extensions and changes that require all players to be using version 3.0.
(for example, the following will still work: template selection, alliances, roads; the ubiquitous timer, city chat, skipping standard messages on the map, [F] in battles with AI, confirmation before quick battles, fixing synchronization issues of scenario settings between the host and joining players, etc.)

3. Yes. Even the standard client will receive a notification (via the chat in the scenario settings).
For example: < HD 3.0 RC7 >: Tournament Extensions - OFF
or < HD 3.0 RC7 >: Improved Gameplay - ON
and it will be displayed in a special (different green) color.
If there is no notification for either of these options, it means the option is not active (equivalent to OFF).
The notification occurs when joining the host.