My disc drive has been acting up as well after the new update.. I posted about it already, though. It'd make a beeping sound (like the laser in the DVD drive) and definitely wasn't normal. Couldn't get Microsoft to tell me what to do (sent them an email)
so I'll just live with it until it breaks. It doesn't seem to affect gameplay, though.
About your issue, it sounds normal to me. On the majority of my consoles the disc will spin for a few seconds after ejecting but does not happen with my 360 (or my PS3, for that matter). You should take note that when you go back to the dashboard the disc
will run to identify what game it is so it will say "Play *game name*" on the dashboard. If you're worried you can always contact Microsoft. But it sounds to me like the console is spinning the disc REALLY fast. Games that do this to you you should install
to your HDD to be safe, that's what I did with Skyrim since my Skyrim disc is scratched up. (not major scratches but still noticeable and doesn't seem to be a problem with my 360 being unable to read it)