OK, apparently installation of SoI works differently, so the instructions I gave you before won't work for it. Sorry about that.
At this point, you should probably delete everything and reinstall the game. Just so you're working from a clean slate and you haven't made any mistakes yet.
What I said before should still work for Wars of Liberty and the base game, but according to SoI's install instructions it needs to be in its own folder in "C:\Program Files (x86)\Steam\steamapps\common".
That means, as soon as you reinstall the game, you should make a duplicate of your "C:\Program Files (x86)\Steam\steamapps\common\Age Of Empires 3" folder, in the same location. You should then have two "Age Of Empires 3" folders in that "common" folder, alongside your other Steam games. Then you can do what I said before in the first of those folders, except make only one copy of the "bin" folder and install Wars of Liberty in that copy. Also don't rename the original "bin" folder, which I said before, as that seems to cause problems. My bad, once again.
Then follow the install instructions for SoI in the duplicate "Age Of Empires 3" folder, which you should rename to avoid confusion (the instructions suggest "Age of Empires III Ori"). Then, if you want to play SoI, it seems like you have to switch the names of the "Age Of Empires 3" and "Age of Empires III Ori" folders. At least that's what I had to do. I have no idea why, but I think it has something to do with paths, so I'm sure there's a workaround. To play TAD or WoL again, just switch the names of those folders again. Basically, whatever mod you want to play, make sure it is installed in the folder currently named "Age Of Empires 3".