I have downloaded HoMM a year ago and now have a new laptop. Downloaded the game again but he doesn't play. For a small moment you see a pictogram in the taskbar below but that is quickly gone.
I tried to solve it myself but i'm afraid i don't know much of computers :)

UPD: Oh wait, forgot something. VERY FIRST thing you should try:
ALT+F4 -> Restart.
Not shutdown, not hibernate - exactly restart.

Common problem for any WindowsNT OS (not a Win10 problem, but still newer Microsoft OS worse and worse). A lot of threads about it. But im never saw solution (100%) and almost never faced that trouble.
First i can suggest you:
reinstall game to X:\Games\HoMnM-3\ -> no spaces, special symbols, cyrillic\umlauts\etc, no Program Files and better not to C: partition.
Enable UAC if you turn if off, and start installer by RMB - run as admin. Set adminrun to game exe. Remove any else compatibility from exe properties (even one set by GOG, experiment with it).

If you start game using GOG Galaxy - set galaxy exe to adminrun (and reinstall it out from program files)

If you have additional antivirus - disable it temporary.

There is system DEP (and CPU one) function that at same time:
- better keep on
- never properly worked in Windows
- never properly choose which one apps should be DEP restricted. Used for most apps, Steam\Galaxy + games as well. But for me it never cause problems, i think.

Set DEP to default (let Windows decide, used mostly for system apps-services (as Windows say, but applied to almost every process))

If you change DEP\UAC - better to restart OS.

Better to use some application as general launcher for games. Steam and\or GOG Galaxy. Set their exe adminrun - do it in additional menu 'to all users on this machine'. Then...
go to Win registry HKLM\Software\Microsoft\WindowsNT\CurrentVersion\AppCpmpatFlags\Layers\ and find that exe file(s).
Their compatibillity should look like this
DISABLETHEMES DISABLEDWM RUNASADMIN DisableNXShowUI DISABLENX
First two - disable themes. One of them is not apply to Win8+, i dont remember which one. Last two - is disabled DEP only for that exe ad not matter what set in DEP settings.

Any else exe\games starting from Steam (or galaxy) get same compatibility\priority\core affinity as parent exe process. You do not need set adminrun to every games exe's.

Or you can run the game through the HD-mod. It solves nearly all HoMM 3 related problems.

Thanks for the help, it works :)

But here is not a HoMM 3 related problem.

HeroesGL mod surpass HD-mod. In case if you need maximum authentic compatibility. Still both is a mod things, not some compatibility-fixes thing.

I have the same problem like OP has. That is what GOG support recommends too.
I Tried that, at least the game is running now BUT only in a small window with much "frame" around it. I tried almost all resolution settings in the Mod, it doesn't start in full screen. That's not the way I'd like to play the game (first time), the window is too small for me to enjoy anything.

Does anyone know a solution? How to get the game work into full screen mode?

The following settings are optimal for me:
Mode: 32 bit True (stretchable) GDI
Source size: 800x600
Stretch filter: Bilinear SharperX2 (M)
And Full Screen Mode and HD+ checked.

But, of course, things may look very different different based on your screen size and video card.

After reading your answer I tried again and found a setting which I can live/play with.
All settings are the same except Source Size, it's 1360 x 768 which works nearly perfect for me (800 x 600 is not possible because of my display I guess).
Thanks a lot for your help PetrusOctavianus. Finally I can start my HoMM3 experience :)

Sehr gut!