As for as editing messages.mo.php, there aren't any secret techniques... you are doing everything correctly.

The problem isn't that your browser is caching the translations, it is that your server is!

I actually struggled with this for quite a while myself, and I eventually made some change to my server that made things easier. It has been a while, I'm sorry but I forget that the change was that worked in my case.

But even now, whenever I update a messages.mo.php file, I have to restart Apache before the changes are live.

I use CentOS, so restarting Apache looks like this:

systemctl restart httpd

Hopefully restarting Apache will be the only thing you need to do. If it isn't, you'll want to investigate what packages you have on your server that create caches. In my case I think it may have been related to PHP (some sort of a cache for PHP scripts or something).

Good luck!

For Example tried to changing the original (line 61)
'Other Resources' => 'Andere Ressourcen',
to
'Other Resources' => 'Wichtige Hinweise',

The Problem is, i never see that change on the frontend. Also deleting Browser Cache etc.... 

Maybe you can help me? Other solution could be, I send you an file with all my desired changes and you 
prepare us an offer with your working hours and price.

Thank you very much.
Andre