Thanks to all for the nice feedback!
To answer a couple of questions:
Hehe, since the axis are skipping if the filament happens to get caught just the slightest, I would leave milling out of the questions.
Well of course you will not be able to mill steel or probably not even any kind of metal. But all the DIY CNC machines rely on pretty stock NEMA17 motors, which are the same as Solidoodle uses. And while I guess leadscrews would be better, quite a few use transmission belts anyway. So in principle the motors should provide enough force for some milling. Maybe you have some low voltage on the trimpots, or the belts are too loose?
The cutter head is at the moment not stiff enough, but I will try to address this problem. The real big problem from a mechanical point of view is the bed, which is not very stiff or robust (and I even have the plywood version!) and it has to raise actively against the bit... It would have been better to have a head that pushes down instead. But we will see.
Have you found any good software to run the arduino based controller as a cnc?
As mentioned above, grbl seems to be the "gold standard" but I want to be able to use Marlin, since I don't want to change the firmware every time. Anyway, I basically only use the movement commands (the bit rotation is controlled by hand). I found this nice script https://bitbucket.org/raw/cnc_mendel/src that converts pycam code into marlin GCode. You will still need to edit it a bit by hand, but it works in principle.
What I tried was to move the bit to the bottom-left of the area where I wanted to engrave, lowered it until it touched the wood and then I run a G92 X0 Y0 Z0 to redefine the "home" there. And then I could let the script run. The engraving failed because of mechanical reason, this will be fixed in the next release cycle 
