The front mis-shift has been fixed, didn't happen again after I setup the front derailleur.

1) I shift to middle/3rd combo, that's right.
2) Front hasn't mis-shifted anymore. Rear still mis-shifts by itself while I am putting force on the pedals.
3) At a slow speed, when shifting and accellerating, it does the very same thing.
4) I have had new wheels installed, with a same 9spd Tiagra cassette as was on the old rear wheel. I got into a crash a while ago, and the RD got bonked onto the road.
5) I will check during my lunch break this afternoon whether it's misaligned.

While riding and looking down towards the RD it somehow looks as if the bottom of the RD comes a slight bit more towards the chainstay than the top. A bit as if it's bent, not entirely sure how to explain what I'm seeing.