Some progress has been made – I managed to launch the game through a virtual 32-bit Windows XP. All the problems mentioned above have disappeared – the TCP/IP button now works. I haven't tried playing through Hamachi yet, but in GameRanger, the next problem is that when I join a game (using the "join" button), the program launches Heroes, stays in it for about half a minute (showing the background image of the menu without buttons), and then kicks me back to the desktop. There are no logs, and GameRanger remains running. I've tried all possible versions, including the one mentioned above. Any ideas on how to fix this?

Added after 2 hours 59 minutes
I figured it out! I was just trying to connect to games that had already started (with the green circle). I can join other games normally, and the game runs very stably – there were no crashes at all during the evening.

After scouring both Russian and English forums, I can confidently say that it is practically impossible to fix the game directly on Windows 8 – install a virtual OS, and all the problems will go away (I personally used Oracle VirtualBox).

Hooray!