- where exactly the three-dimensional surface got touched (the main problem!).
That does sound like the main problem to me.
The only solution to this problem that I know of is skin, either nerves in skin or hairs attached to nerves in skin.
You should define the size of your object and the resolution of touch you want to detect.
My gut feeling is that flexible conductive hairs would offer the best solution, but how you would fabricate something like that I have no idea.