Do harddrives slow down with age?

No, harddrives don't get measurably slower with age. Drives can get worn mechanically, and they can get occasional bad sectors, but either they work for decades or they fail hard and quick after a while - not a slow decay. As Ignacio states, there's a bit of age-related maintenance inside the drive, but that's on a scale you wouldn't notice.

Windows is known to slow down (see, it's software-related) over time, especially if you install&uninstall applications often. At any rate, if the machine is running for 6 years on the same Windows installation, you're doing well! I would suggest to back up the machine, then reinstall the OS and your programs.

I agree with you that the specs of the machine make it useful for several more years. If you have (access to) GRC.com's SpinRite, you might want to try it out. It can refresh your disk.

As hard drives age, they may need to remap reserve sectors in place of non-working sectors. A SMART tool should be able to tell you how many of these remappings have been performed, as well as other factors that may be a result of old age.