Thanks for you help Falko!
First, I now can receive mail. I reinstalled the site and got the pop3 with webmail started.
Still no sending unfortunately
I checked the blacklist, but I was only listed in the SORBS-DUHL-list, which as far as I understand is a list for adresses in a DHCP-span. The other lists where flagged ok.
Basically when i check var/log/mail i get the "Jan 24 05:21:16 franke postfix/smtp: connect to alt1.gmail-smtp-in.l.google.com[188.8.131.52]:25: Connection timed out"
"Jan 24 05:55:58 franke postfix/smtp: connect to mx2.hotmail.com[184.108.40.206]:25: No route to host"
I begin to get desperate here and really appreciate your help.