How do I access POP/IMAP/SMTP through a proxy? (The proxy settings in Thunderbird do not work.)
The same way you'd do it for any program not supporting proxy - with a 'socksification' tool. However, because you use a HTTP proxy, rather than SOCKS, this is in fact very complicated.
If your HTTP proxy allows CONNECT command to your mail servers, then you can use a socksification tool directly with that. However, I have not found a free Windows socksification tool that supports HTTP proxies; ProxyCap and SocksChain do, but are not free.
If, on the other hand, your HTTP proxy does not allow CONNECT to your mail servers, the only way to get around is via an intermediate server you have access to. This is not easy to set up, and there are multiple ways of doing it. If you don't have networking experience, you probably won't succeed. To put it simply, you'd have to:
Usually a proxy is used to control the access only to the web (http on port 80 and https on 443). Hence you shouldn't need to configure the proxy to access any of POP/IMAP/SMTP. The proxy setting in Thunderbird will be used only when reading an email that has http content linked to it.
I would recommend that you check if secure access is mandatory for your POP/IMAP/SMTP servers and make sure to check the corresponding options such as SSL, etc.