I'll describe it now, but first, a moment of admiration for the person who hasn't followed the Heroes topic for 5-7 years.

Many HD users received all the features gradually, over 9 years of development.
If all the current functionality was released at once, it would be a bombshell, yes.

By the way, there's HoMM III with increased resolution and the title "Remastered" on Steam... But no, there was no limit to the disappointment - it's just SoD, just a different resolution.

Correction: just RoE.

So, about the errors...

It would be logical for the update program to check if the launcher is running, close it if necessary, apply the update, and launch the new version, as it generally happens in the normal user world.

Actually, that's how it happens with the HD update for most users. That's why I asked about the errors. It's a pity that the description is unclear and not very detailed, but judging by it, the problem is with access rights to the folder and files in it.
Heroes were created when Windows 95 and 98 were in vogue, and users had unlimited rights in the system.
Today, Heroes and the HD mod need to be installed in a folder to which you have full access.
Such a folder is NOT: System drive\Program Files (x86)\...
Installing not in Program Files doesn't always solve the problem if your Windows account has some additional restrictions. Plus, an antivirus program may interfere with HD.

Another thing. If you click on the shortcut, the game starts (let's say, in full screen) and a separate window of the launcher reports that an update is available. This is, of course, cool, but it would be logical to check for updates before launching the game itself, update itself, and then launch the game itself. Otherwise, there is simply no point in the automation process, because you have to close the game, click "update", close the launcher manually, update, open the launcher, make sure there is a new version, and play.

1. The update is checked in parallel with the game launch, simultaneously with it. If you do it the way you suggest, first check the update and then launch the game, the game launch time will increase.
Depending on the state of the network from you to the update servers and the state of the servers, the launch time can increase significantly. At the same time, why should the player wait for the results of the update check if there are no updates, i.e., the latest version is installed.
2. Usually, you don't need to close the game or launcher manually. When you click "Yes" in the update window, HD closes the game and launcher itself if nothing (Windows) prevents it.

specifically, I'm interested in, for example, whether it's possible to enable this:
https://sites.google.com/site/heroes3hd/rus/description/extended-ui
... Found:
UI.AdvMgr.SkipMapMsgs
It's strange that it's not enabled by default when it's in the description...

Actually, it's enabled by default. It seems that in this case, HD again has a problem with accessing its files.
But if anything, it can be enabled/disabled in the game: system options -> show object messages.

- Sys.CPU.ReduceUsage - Reduce CPU usage? How does it work?

set the value to 1 - the CPU load decreases.
If, at the same time, mouse lag appears (it will appear if you have a mouse with a high polling rate, some kind of gaming mouse) or some other lag, then set the value to 2
if the lag still exists, then return to 0.

you can also reduce the CPU load by choosing a lighter graphics filter in the launcher: for example, the one that is set by default: Bilinear SharperX2

***

The description of HD features on the website has not been updated since January 2016. The information there is not complete and is sometimes outdated. So, you may not have discovered and tried all the HD features yet.
Unfortunately, I don't have enough time for everything.
It's hard for me alone to support and develop the mod itself (> 100,000 lines of code in it). And I don't have time or energy for documentation, feedback, and PR.
You're lucky, I answered.