I'm the Disco Yeti expert so I have answers to your questions.
Expedition Everest opened in January 2006 for the soft opening and the grand opening was April 2006.
The Yeti was fully functional through the end of 2006 but then it went into "Disco" mode.
There are people who claim to have seen the Yeti working as recently as 2008 so it's possible Disney tried to make it work a few times after they originally shut it down.
The Yeti has definitely not operated within the past 5 years (2009-2014).
The Yeti is fine, the problem is the structure the Yeti is standing on can't handle the force the moving Yeti puts on this structure (I think it's concrete, but not 100% positive). This structure supporting the Yeti has cracked and the big fear is that the Yeti would topple over onto the track and cause massive injury to passengers, most likely death to an entire train of people. Therefore Disney just can't take the risk of operating the Yeti until it's foundation is fixed.
Now, Expedition Everest is actually 3 separate structures. The Yeti, the Coaster, and the Mountain are all independent of each other. To fix the Yeti Disney has to remove the track around the mountain and part of the mountain itself to even get to the Yeti. Time estimates are approximately 9 months to remove and rebuild the track and mountain. So you're looking at EE being closed for 9 months.
Disney will fix the Yeti as soon as Animal Kingdom can operate successfully without Expedition Everest. The best estimate is that once Avatar land is complete Disney will probably schedule a time to shut down Everest between Sept and June so it won't miss a Summer season. My best guess is 2020 or later.