OK, I googled it (didn't know exactly what to google at the time) and found the solution.
I downgraded the API Level to 17 just to be sure (Android version 4.2.2) and it works like a charm, both in the emulator and on my physical phone. :-)
Thanks for the help. :-)
A question though: Do I even need the API Level 20 stuff? Or is it safe to just uninstall them and just work with Level 17? Because the whole thing takes a billion trillion (rough estimate) Gigabytes on my HDD and it's a bit annoying since it's almost full... :-/