Nope - you can make a capacitor out of any shaped materials - consider two bi-concave lenses facing each other.
It will depend on the details of how the surface is arranged. You'll have to do the math.
Most complicated surfaces can be understood in terms of a network of regular capacitors.
Yes you can. The simplest example, I think, would be wrapping a regular parallel plate cap into a spiral.
A network of capacitors, in other words.
How do you have to join two capacitors together to increase the overall capacitance?
But I think you should be using, as your basic model of a capacitor, something involving paired
conductors ... one providing charge and the other receiving it. The concept of capacitance does not really make sense if there is only one conductor.