I think falko did with the bit highlighted...
May 17 22:55:00 server postfix/smtpd: warning: connect to mysql server 127.0.0.1: Can't connect to MySQL server on '127.0.0.1' (111)
May 17 22:55:00 server postfix/smtpd: warning: mysql:/etc/postfix/mysql-virtual_client.cf: table lookup problem
It looks like your mailer cannot confirm virtual recipients configured within ispconfig as it cannot gain correct access to mysql.
Check /etc/postfix/mysql-virtual_client.cf see if its got the right username/password/database name in etc for access to ispconfig tables.
Perhaps install a copy of phpmyadmin, gain access to the databases, see if they are populated properly.
Check where you are expecting your email to be sent FROM and TO, both email addresses and ip addresses, as this part looks a bit wrong to me:
May 17 22:55:00 server postfix/smtpd: NOQUEUE: reject: RCPT from localhost[127.0.0.1]: 451 4.3.5 <localhost[127.0.0.1]>: Client host rejected: Server configuration error; from=<office@eXpert-2002> to=<firstname.lastname@example.org> proto=ESMTP helo=<example.com>
So, a submission from 127.0.0.1 sending an email with NON-FQDN from name and a possibly invalid HELO name unless you hid the actual name, but if that's the format may still cause a rejection the HELO itself doesnt appear to be FQDN. RFC2821 is your friend.
Have a look in mailer logs again, are you seeing ANY mail at all eg server messages to root, unless you've redirected them see /var/spool/mail/root.
Is MYSQL definitely running and can your mailer definitely get the correct access to it?