Thomas T. Frost wrote:
I work with memoQ servers all the time. There is no problem. I have version 10.3.10 right now, and I guess some clients may still be using 9.x servers. It's not a problem. I did encounter problems many years ago while using v7 with a v8.x server, but this is not relevant today. Some clients mistakenly think you have to use exactly the same version as their server, but you can just ignore that and rely on the version compatibility policy memoQ support mentioned.
Thank you very much, Thomas! You have helped me find the best way to proceed. 
You haven't explained anything about an 'old server'. If a set of server credentials are no longer needed, you can tell memoQ to forget them from within a project: Project tab > memoQ TMS > Manage logins. It's a bit hidden away.
This simple solution did not stop the error messages from popping up periodically. Fortunately, memoQ Support kept offering various solutions until the issue was finally resolved.
But you are worrying too much. Just use your up-to-date version to check out projects from their server, and you'll be fine. No need to discuss it with them, as they are clearly ill informed about version compatibility. You'll just worry them.
I have already informed my client, as I believe they can also benefit from better understanding version compatibility.
Thanks again,
Ildikó