Sims 3: Don't mix CD and download
After several chats with people from customer support (thanks for your endless patience!), I found out why my brandnew Black Friday offer Sims 3 expansion pack didn't work.
Download stopped at 96%, a pop-up telling me "Oops, something went wrong, try again later". Nothing from the list of trouble shooting worked - exceptions in my anti-virus, re-installing the EA app and so on. Every single hint from the internet was misleading. But it worked on another PC, so I had an idea. My original Sims 3 and several expansions were installed from CD and the new one was the first download.
Long story short: I uninstalled Sims 3 and installed it again via EA app. It works just fine. So, if you experience trouble with a new expansion and haven't played in a while, think about how it was installed at some point. Apparently CD and download don't mix.