“Best map for Heroes of Might and Magic III” (2007).

What is the map's theme? Or can it be absolutely anything (any theme)? And also, is Dungeon very important?

The theme is up to you, meaning it can be anything! ;)

Is it mandatory for the map to include a hero with the nickname of a forum user?...Is this very important?...

I don't think it will be very difficult to do, but by doing so, it will immediately be clear that the map is personally yours, and not taken from another online resource (of course, I don't rule out the possibility of redoing the map, but I still think that it is unlikely that anyone will do that).

1. Are there any restrictions on races? Artifacts?
2. What is the minimum size of the player's starting army?
3. Is the maximum possible income for a player limited (taking into account ALL mines on the map and ALL cities at the highest level)?
4. And the theme:)...

1. No.
2. Up to you, meaning anything.
3. No.
4. Anything.

But, you need to remember that the map should be balanced and playable, and also difficult to complete. Such a map will have every chance of winning! ;)

Sincerely, VDV_forever