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.

Conventional rotating drives may not slow down with age, but solid state drives (SSDs) may slow down with use (especially when they're filled to near capacity). This is more apparent when dealing with older SSDs, older OSes, and/or older drivers that may not understand or fully utilize facilities that combat this, such as TRIM.

See AnandTech's "SSD Anthology" and other articles for a detailed description of why: essentially, SSDs have different limitations than hard drives (e.g. their "erase block" size) and to act like traditional hard drives they need to have a "translation layer" of sorts.

That layer ends up being almost as complex as a modern filesystem.

Filesystems themselves can experience performance degradation in a number of similar cases:

• large numbers of files or directories
• fragmentation (as you pointed out, a defrag helps with this)
• low remaining space (it may take more "work" to determine where to place pieces of files)

All that having been said, I agree with torbengb: your issue is much more likely a software one.

A few options:

• If you feel up to it, grab the Sysinternals Suite or similar tools and do some sleuthing: it may be a single app, dll, or service that's hogging some critical resource (disk, memory, or CPU). With luck you can just disable or reinstall whatever is being pesky.
• Boot from a live CD or USB stick, or install a fresh copy of Windows on a separate partition. The USB or CD may not have the same performance characteristics, but if things feel zippy again, it's likely software.
• Bite the bullet: back up, format, and reinstall, then be selective and take time when putting apps back. Most power users end up doing this every so often anyway, and it almost always provides a tangible result.