Is my hard drive starting to fail?
There is no harm in getting a call logged before your apple care expires, although from your description nothing you say would suggest to me that the hard drive is at fault.  It sounds more like the slow build up of unwanted startup items, and inevitable disk fragmentation is just slowing things down, or you could plain be out of memory more often than before.  If you have a secure backup, I might consider a couple of things to test:

1) Create a new user, and see if it is as slow as your usual user account
2) From a complete backup, format your drive and restore to defrag the drive, and again see if the same happens.

HD problems would usually manifest themselves in unusual noises, clicks and whirs, repeated attempts to load a file, genuine errors in log files where you could use the Console app to search for things like "disk0s" "error" and "I/O error" - especially if they correlate in time to the slowness or freezing you are seeing.