One of the main features of Safe Mode is that it disables your Extensions. In reviewing your "More system details" I can't help noticing that one of your Extensions is named Flashblock. Is it possible that you need to add an exception for the site where Yahoo videos are hosted so they are not blocked?

You were on the right track. However, It was the No-Script Extension that was causing the problem. I went to the No-Script Options then General tab and checked the box "Allow top-level sites by default".

I had noticed  that as I kept trying to play the videos, a No-Script message popped up about something (don't remember exactly what it said). 

The Yahoo videos now play.