Is Heroes 3 a sport, like chess? How does the planning differ?

So, in chess, if I understand you correctly, when playing online, one player can use computer assistance, and if the second player does not use this assistance, then the first player gains an advantage.

This isn't even an advantage; it's an inevitable win against an opponent of any level.

And now, here's the question: are you aware of any cases where someone seriously calls online chess a sport?

I was actually referring to playing in person, not online. Online platforms have anti-computer protection. The system scans each game for the use of computer assistance and identifies it with high accuracy. I don't know how it does it, but it works, and the accuracy is almost 100%. There is no such protection in person, even in serious tournaments.

Are there online tournaments where the rules prohibit the use of computer assistance, and where there are penalties for doing so?

Such actions are prohibited on any online chess platform. The rules provide for a ban as a penalty. However, there is an option to enable "using computer assistance" and play with other players who do the same; some people find this interesting.

Are there any serious online competitions with prize pools comparable to those possible in chess today?

Serious prize money is currently available in show tournaments, such as SpeedChess, where the prizes are in the tens of thousands of dollars (if I'm not mistaken), but this is only for the elite.

Added 1 minute ago

In chess, planning is constantly ongoing, while in Heroes, there is a wait for the opponent's turn.

In chess, as in Heroes, people also think during their opponent's turn! Imagine!!!!