Hi Stefan,
I hope that it doesn’t sound like a cop-out, but it is definitely worth mentioning that I doubt that this issue is related to osTicket Awesome.
Assuming that your installation is in a directory called support, if you followed the installation instructions then your original osTicket installation will now be in a directory named support-old.
Please visit your original osTicket installation in your web browser and see what happens if you upload/ download attachments from there.
If the issue persists while you are using the original interface, I would recommend seeking help for this issue in the main osTicket Forums.
But if it does turn out that the issue is specific to your osTicket Awesome installation, do let me know and I will give it some more thought.