You have a couple errors:
PHP warning: stream_socket_client(): unable to connect to IPADDRESS
Since the rest of the error appears to be German I think that it basically says that "the connection failed because the remote site did not respond in time". This could be a firewall, or a software firewall. This could be a networking issue. This could be a port issue. This could be that the mailserver you are trying to connect to does not allow connections from the webserver.
Which Ports must be open? 25, 465, 993?
Whatever port the mail server wants you to connect with. Which of course depends on how you are trying to connect to the mail server and how the mail server is configured and if you are using SSL, TLS, etc. This isn't something that we can really tell you. You should talk to servertown.ch (your webhost) for answers to these questions.
Your second error message says
Failed to connect to mailserver localhost.
Are you running a mailserver on your web server? If not then I presume that osTicket is trying to use PHPmail to send emails (maybe because of the previous error, or maybe because you changed the setup to phpmail).