OK, I've uninstalled Avast using the clean uninstall tool. I've run the McAfee clean uninstall tool again. And now I'm running a Malwarebytes scan. Oddly enough, there doesn't appear to be a clean uninstall tool for Comodo Antivirus. Hopefully that's not what's causing the problem. Anyway, if the malware scan comes back clear, I'll reinstall Avast and see how I get on.
Also, on a side note: one of the first suggestions I got in this thread was to "exclude" Avast and Comodo Firewall from each other, but no one ever told me how to do that. How do I do it?