It's totally normal to get a wrist watch for a few seconds since your computer is scanning for bootable volumes. The odd thing is, you should be able to move the mouse and select a disk afterwards.
Try a different method:
Start your computer, insert disk and go to System Preferences -> Startup Disk, select Hardware Test + Restart.
Running the Hardware Test to identify any potential logic board failures seems to be a good idea, though it can only detect like 99% of all hardware issues.
Also, the fan starts a few seconds after power on, but runs for only about 2 seconds before stopping (and it does not restart at any point).
You shouldn't hear it unless you're performing some CPU intensive tasks, so it does not necessarily indicate a problem. You could use
Temperature Monitor to check for any unusual temperatures. I've got around 50°C without any fan activity.
Hardware Test is the only diagnostic software available as far as I know, besides various hard drive tools. A technician would either run Hardware Test or open your machine and measure some individual circuits.