If I can't use the Lotus Notes client, can I use another third-party email client (Outlook, Mail.app, Thunderbird, etc.) to connect to a Domino server (either through plugins or a built-in option)?
So from my current understanding, it seems that Microsoft Outlook with an additional "add-on" is going to be the only other option besides Lotus Notes. The steps to install the additional add-on can be found on e-how.com.
Domino supports POP3, SMTP, IMAP.
Your question is unclear and the Edit appears to have the answer in it. The 4 clients mentioned all should be able to connect to the server if you have those services set up.
You can install a plugin in outlook to allow access to some domino servers. I'm not sure that it gives full access to everything or not. The best bet would be to get the Notes admins to install Lotus Domino Access for Microsoft Outlook (DAMO) or to see if they will enable imap and pop3 access on the servers. If the admins aren't open to modifying the server config, you really should just use a Notes client. They make one for basically every OS so there isn't an excuse not too and you aren't going to have full functionality otherwise. None of the plugins are going to give you access to any of the domino applications outside of mail and calendaring and most companies that use notes also have a ton of application databases as well.