Re: Exe files do not exist

C:\Program Files (x86)\Origin Games

This should be the default folder for games installs.  If you have changed this option under Origin options, then it could be somewhere else, aka, another drive, or folder. 

In origin games, you should have a DAI folder, and the .exe should be in there.  You may try a search of your computer for DragonAgeInquisition and see what pops.