HoMM3 HD is a mod (add-on) for the game Heroes of Might and Magic 3.

R55Max,
It's hard to understand what you're saying; basically, nothing is clear.

After installing the HD mod and performing some manipulations, possibly incorrect ones, the game stopped launching in fullscreen mode.

So, it only launches in windowed mode? And what happens when you press F4?

When I installed it the first time, there was an option to change the resolution, which I did, and it changed and now looks like this.

And how does it look now? How did it look before?

The maximum possible resolution was 1024*768, even if I set a higher resolution in the mod settings.

So, even if you set 1280x800 (the physical resolution of your screen), the game still launches at 1024x768? If this is really the case, it's the most unbelievable thing you've described.

I decided to revert to the previous resolution, but after further changes, the game no longer launched in fullscreen mode. The resolution did change, but it was smaller than the entire screen. The game image was simply scaled up or down relative to the screen on a black background.

So, when you set, say, 800x600 (aspect ratio: 4:3) or 1024x768 (4:3),
the resolution is set while maintaining the aspect ratio, but you want it to be stretched to 8:5? Well, many modern monitors and graphics cards allow you to configure how a resolution that doesn't match the aspect ratio of the monitor's physical resolution should be displayed. And if your laptop used to stretch 800x600 to fill the entire screen, but now it displays it with the aspect ratio preserved and black bars, it's obvious that your laptop has this functionality. Therefore, instead of uselessly reinstalling the game and HD mod, you should try to tinker with the image output settings on your laptop (either through a special proprietary utility or simply through the graphics card settings).
As a last resort, you can try reinstalling the video drivers.

In general, there's no need to panic, and the HD mod is not the problem. The HD mod uses a completely standard method (just like the original Heroes) to send a command to DirectX to set a specific resolution (from the resolutions supported by the system), and then it's completely unrelated to the HD mod how DirectX processes this command, how it passes it to the graphics card, and how the graphics card ultimately reacts to it.