When playing versions 346 and 348, synchronization issues do occur...

Even in 3.47 HD+, I needed to make the network incompatible with earlier versions because the battle calculations had changed. But I forgot.
In 3.48 HD+, I needed to make it incompatible with previous versions again (because the battle calculations had changed compared to 347 and 346). But I didn't bother.
I'll do it in the next version.

If they could also fix the display of heroes in the tavern on the first day of the game. It's a really bad thing.

I don't understand what you're talking about. I guess I'm not aware.

And also, not so long ago, some guy asked me if I knew how to remove the black bars on the sides in the HD mod, so that the resolution would be like in the original Heroes, filling the entire screen. I didn't know what to tell him. Is this even possible?

If he has bars in 16-bit, 32-bit GDI, or 32-bit True Color modes,
then he either needs to choose a game resolution equal to the physical resolution of the monitor, or he needs to tinker with the video card settings.
But in this case, there should be bars even with the original game.

If he has bars in 32-bit MMX (stretchable) or 32-bit True Color (stretchable) modes, then this means that the game has a resolution selected with an aspect ratio that is not equal to the aspect ratio of the desktop resolution. The launcher offers non-standard resolutions with an aspect ratio that matches the screen for selection.
In these modes, bars appear when the aspect ratios do not match, which is inevitable, as the image in the game is stretched PROPORTIONALLY.
All video cards (if using default settings) stretch the original 800x600 (4:3) to the required size, for example, for widescreen monitors 16:9, NON-proportionally to fill the entire screen, which distorts the image by stretching it horizontally – that's BAD!

The beauty of (stretchable) modes is that you can choose a resolution smaller than the physical one without distorting the image in full-screen mode.

It is not necessary to use the proportional resolutions offered by the launcher; you can calculate your optimal resolution using the proportion: video lesson (mathematics, 6th grade).
If you need a flattened, distorted image with a non-standard resolution, then HD won't help.

Well, here's another bug: it's when in a computer battle, artifacts are transferred to the loser during teleportation!

Actually, this is not present in HD+.