You would need to get straight in your own head what you mean by "cannot send or receive email" - that covers a multitude!!
- Have you tested from your machine to another mailbox on the same machine?
- Are your MX records correct and current?
- From the mail logs it appears your dovecot is working and you can log into it, do you have squirrelmail or roundcube or similar installed, what happens when you submit a mail from there? Is it accepted, is it rejected?
Parts of the log that jump out are:
May 23 03:43:50 server1 postfix/qmgr: 9D61D1A2259: from=<root@XXXXXX.co>, size=1616, nrcpt=1 (queue active)
May 23 03:43:50 server1 postfix/qmgr: 948851A2204: from=<root@XXXXXX.co>, size=1616, nrcpt=1 (queue active)
See the "from" part - xxx.co. Does the FQDN of the machine acutally end in just .co or have you simply masked that part. An incorrect FQDN can cause hotmail to refuse connections, as well as incorrect reverse DNS.
See there's connections timed out to hotmail those mails should be queued - do a "postqueue -p" and see what it says for the reason. Can you telnet manually on port 25 to any of those addresses, do you have any firewall in the way that could be blocking you.
Seems to me the mail's being submitted to postfix but its unable to relay on.