Works fine by using Proton 4.11-13 or another version with PROTON_USE_WINED3D=1 flag. After launch will be minimized to task panel (or dock), so just open it again from it.

For disable intro delete or rename file ".steam/steam/steamapps/common/Heroes of Might and Magic 5/video/intro.xml".

You may want launch windowed game. For that set in user.cfg for your profile that located in folder ".steam/steam/steamapps/compatdata/15170/pfx/drive_c/users/steamuser/My Documents/My Games/Heroes of Might and Magic V" this variables as example:

setvar glx_fullscreen = 0

setvar glx_resolution = 1600x1200 (or another)

setvar window_pos_x = (monitor-width - 1600) / 2

setvar window_pos_y = (monitor-height - 1200) / 2

Don't use PROTON_USE_D9VK=1 or latest Proton without PROTON_USE_WINED3D=1. This will result in missing textures (need to rechange resolution in game) and campaign won't start.

Slightly stretched proportions in 16:9, and don't playable in 21:9 because of this.

Performance much lower than on Windows, but still playable.