I changed the password for my Google account earlier, and since then, every attempt to install an app results in a download was unsuccessful error. I have not been prompted for my new password anywhere, so this is highly likely to be the problem. How do I fix this?
You cannot remove the "main" account that is linked to your market account, at least I have never been able to when I tried.
I would try rebooting the device, then turn off all "syncing" for that account. I would even reboot again after turning it off. Then turn syncing back on.
Depending on your version, try going into home -> menu -> settings -> Accounts & sync.
You may need to remove and re-add your Google login as I don't see any option to change the password (in v2.3).