How can I test if my hard drive is going bad?
When i am having all sorts of issues like you describe, i try booting up from Disk Warrior and running its tests.

From your description of symptoms I would suggest that it is a possibility your laptop could have directory, memory/RAM or even at its most extreme, motherboard issues.

Apple's Disk Utility only goes so far. Oftentimes the issues it discovers are caused by other problems from the machine. The best indicators I have seen to an HD failing involve files not being able to write to the disk, or files becoming corrupted. Also, if parts of the System files become corrupted, you would see finder and other system issues ... Like you described. A previous poster's suggestion of booting from another drive is a good one to eliminate this as a possibility. (I use an external HD instead of a memory stick.) 

The absolute best safe guard against losing data on a HD is regular back-ups.