.... and the big question is, has your disk actually run out of space? Have you checked?

Have you read the wiki article (or any of the many forum threads) on how to configure a relay for zimbra?

It's a firewall or DNS problem and nothing to do with zimbra, do an internet search for the error message.

Does your ISP block outbound port 25? As I said, there is a 'block' somewhere and there's nothing wrong with those servers, I can telnet to port 25 on any of them even on my mobile phone.

Conection timeout would indicate a possible network or firewall issue.

Have you looked at any of the wiki articles or forum threads on the topic of 'backup'? I'd suggest you do that and try some of the solutions but the important part of this is a working backup and restore.

I do have ZeXtras but I don't use chat so I can't comment about that. You can install ZeXtras as a thirty day trial to see if there's a problem or not.

There should be no problems doing that or you could always upgrade to the most recent ZCS version on your current server then move that to your new server. Don't forget that you'd need to have a working backup and restore procedure in place before doing any of this, problems do happen from time to t...

What you should do is install the most recent ZCS version (currently 8.8.15) on the new server then use the ZeXtras Migration Tool to move all your settings & mail etc. to the new server.

You're not allowed to change the Zimbra logo on the OSS version:

https://blog.zimbra.com/2015/09/change-login-app-logo-open-source-network-edition/

Make sure you follow the licensing guidelines.