This shows both the Live and PTR. Even though ROC and TFT are listed separately, they both link the same installer.
Install "blizzard-battlenet-standard" https://lutris.net/games/battlenet/. Until reforged, can't directly use the real battle.net installer, but the Warcraft 3 Classic variant of the battle.net agent has the same dependencies.
The remaining text assumes the default
~/Games/battlenet/ wineprefix location that this creates; adjust this for your configuration accordingly.
Battle.net.exe app prior-to/concurrently-with
This is the tricky/broken part. It seems blizzard has bolted-on a buggy separate battle.net launcher made bespoke for WC3C.
Trying to run Warcraft-III-Setup.exe standalone results in an error message linking to the completely-useless information-devoid article https://us.battle.net/support/en/article/19031 for error code BLZBNTBTS0000005C.
The actual issue from the logs appears to be related to a bug in Battle.net Agent build 6700's self-updating feature.
A reproducible workaround is to have the battle.net installed previously by lutris running/logged in prior to starting the installer. Presumably IPC is used to the running agent, which skips the bugged self-update process.
When you start the installed
Warcraft III.exe, it will create a completely blank grey window. This is also a battle.net agent bug. Drag the window out-of-bounds and back in to view, and it will redraw. You can do this continously as you type your battle.net credentials.
Once logged in, you don't need to do this again.
black screen/broken resolution switching
I use xmonad, which cooperates best with what is reported in-game as "Windowed Fullscreen".
-nativefullscr in Arguments seems to apply this.
You might also try
-window just to get the game running so you can get to the video options settings.
The current default wine-4.0-tkg is too old and displays black boxes and other artifacts;
wine bugs with the world editor were almost-fixed some time near 4.8.