Supposedly, I have five devices associated with this account. However, the list on the "Manage Devices" page only shows one. I think the others have long since been junked and I would like to remove them from the account.

iTunes authorized devices are not the same as the associated devices listed on Manage Devices. See: Back up your Mac with Time Machine - Apple Support

To remove authorized devices, use Deauthorize All and then reauthorize any computers you still have and want to use. Note that only computers are authorized devices. See instructions below from Authorize or deauthorize your computer for iTunes Store purchases - Apple Support

To deauthorize a computer that you don't have access to, you have to deauthorize all of your computers. Then authorize each computer that you have again. You can deauthorize all computers only once a year.

1. Open iTunes.
2. Choose Account > View My Account.
3. Sign in with your Apple ID and password. Forgot your Apple ID?
4. At the top of your Account Information page, go to the Apple ID Summary section.
5. Next to Computer Authorizations, click Deauthorize All.
6. Authorize each computer that you use again.