Run in compatibility mode for windows 98 on the main EXE. This fixed all issues for me.
Additionally, run setup.exe and make sure the game is not running in maximum performance mode, run it in compatibility mode instead. Maximum performance mode on modern systems seems to break some functionality in singleplayer and multiplayer.

The best option for multiplayer remains to be RADMIN VPN. Hamachi vpn introduces errors and glitches, if you can even get it to work anymore.

Gameranger does not work with version 1.10 of Tzar, this is a known issue and has been reported many times on the gameranger forums to no avail.

And a little secret that I found out as a child: if you rename the campaign editor and other EXE's to something else, you gain the ability to modify and edit the original campaign map files. You can then make them multiplayer or playable in custom games by editing their settings. Have fun!