For the mechanism, as I am no engineer and have no electronics background, I wanted to use a complete solution which would need as little modification as possible.
This is one of the reasons why I did not go with the Arduino solution.
I found out that an RC toy using two engines as its movement controllers would be perfect (I was thinking about a tank on caterpillars). I could not find any (all of them were running on 4 wheels with “steering wheels” in front while I needed steering based on the left/right differentials).
I eventually found an excavator on caterpillars which were running with the mechanism I was looking for and with extra, rotating cabin! I now had the engine for rotation of the BB-8’s head.
There is also a 4th engine for the movement of the shovel which I could use for the up/down head movement but I think there will not be enough space inside of the body to fix it and I do not have any idea how to design that mechanism as yet.
For now, I will just stick to the simple movement of the body and rotation of the head.