I know I'm a "0 Post Wonder" and what not, but I've frequented these forums quite often in the past.
Like so many other people in this thread, I wished to play Warcraft 3 TFT on (Mountain) Lion. Fortunately, it is possible!
This method is extremely simple, but does require two computers; one of which is running OS 10.6 or older. You follow the instructions previously, where you install (through either CD or from us.battle.net -> Account -> Game Clients) AND PATCH both Reign of Chaos and The Frozen Throne by connecting to Battle.net at least once.
----To make things more simple later, please copy/paste the Warcraft lll file in your Applications folder to your Desktop.-----
After that, you can go to Apple->System Preferences -> Sharing. Once there, enable fire sharing on both the Mac with Lion and the one with and older OS.
Shot at 2012-07-27
Now, go to finder, then to the right side there should be an option under SHARED, listing the name of the Mac you plan on transferring from. In the upper Right, there will be a box named "Connect As..."
Manually enter the name and the password of the !Mac with the Warcraft lll file! This is important, as entering the information for the computer you plan to switch to accomplishes nothing. After that, hit the connect option.
In Finder, you should be able to view all files of the User. Either use the search bar or navigate to the file with the Username of the User (usually just "xxxxx"), where x is the User's name.
In that file, choose desktop and in that window, choose the file named Warcraft lll. Drag that file into the applications folder of your Main computer and wait for it to copy. The file is around 2.3gb and should take 10 or so seconds to a minute to copy, depending on your Internet connection.
(Note, it is important to not have a copy of WC3 installed before you do this transfer. I did this the first time I attempted and after working it went back to having the white crossed out circle logo). After repeating everything and connecting to Battle.Net on the new account on the new computer it worked fine. I also saved a game, just to make sure.
Just some proof, so help validate my claim.
Shot at 2012-07-27
(There is no Windowed mode that I'm aware of, so please forgive me for not having a more direct way of showing you I'm playing.