Please offer some advice.

Okay, you can download version 3.51 (or Equilibris) here: http://equilibris.celestialheavens.com/rus/downloads.html

(if possible, with a detailed installation description)

There's nothing particularly complicated about it. Download the installer to the game's folder and run it. Then, follow the same steps as you would when installing any other program. You don't really need to install the TE version.

I just downloaded Equilibris, installed it, but it doesn't work... During the installation, it said: "Warning! The checksum of your heroes4.exe does not match the reference checksum"... what should I do?

Download the files through the same link I provided above. You definitely need to download the .exe file, and then proceed based on the circumstances.

Can I install the Equilibris add-on directly on the first version of Heroes IV, or should I first install The Coming of the Storm, then Winds of War, and then Equilibris?

You can install it, but problems might arise, like the one you described.

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shnurov_hl, did you copy and paste from their forum's FAQ?