The folder is a hidden one, and the %localappdata% is a shortcut to the c:\users\{username}\AppData\Local folder. Typing that into the address bar (or the run prompt) should open it up though.
Support should be able to sort you out, and probably easier/faster than me, the ideas are in case you want to give it a shot yourself.

I found it and the whole thing is filled with this:
Inner exception:
Message: Die zugrunde liegende Verbindung wurde geschlossen: Die Verbindung wurde unerwartet getrennt.."

I don't know why it's in German since the rest is English, but it means:
"The underlying connection has been closed: The connection has been cancelled unexpectedly.."

It says so every couple lines. Nothing I can do it seems.

It's off topic, but the reason is that it's an OS message and you use German as the OS language.

Since you've already contacted support, you must have a support ticket number now. Can you try to send that log file via the GOG Downloader to support and put your support ticket number into the box that asks you to write a message?
To send the report from within the GOG Downloader, click on Settings and at the bottom of the new window there's a button called "Report"; click on that and another window will open, put your support ticket details there and hit "send log file".

I've done that now, thanks for the tip.

And @Vanchann: that does explain it, thanks :)

The 500 http error indicates a server problem. Try again in a couple of hours, and if it's not fixed, GOG staff should take a look at it.