I've been having the "Open Tray" issues recently, most of the time when it does come up I just open and close the tray and it will work. Every once in a while it will say "Play DVD" when I have a game disc, if I press A to play it, it will just hang on a white screen with the 360.
So, when I try to clear the system cache the progress bar will pop up, and then dissapear without filling. I'm assuming it just fails since It never makes me re-download any game updates or anything like that when I do it.
As far as removing the hard drive, I can't say for certain if that fixes the problem or not since the open tray issue will randomly pop up.