2D DIY capacitive panel
My question is, can I use single sheet of copper (or tape), with multiple connection points, to make a single panel, where Arduino will calculate the coordinates from the multiple points? 


No. Due to the high conductivity of copper it will not be possible to differentiate between different points on the surface.


  It's in case I want to change the layout, the only thing I would need to do is replace one sheet of paper and some layout file for the Pi (it will decide what to send according to the layout).


To do this you would need to create a matrix of pads with sufficient "pixels" for your most demanding application.


  I've seen some Instructables for single-dimensional sliders, where they connect each end of the slider to a pin, so I think it will be possible.


I haven't and you haven't provided a link.


  How many points would be ideal? Could I just triangulate it from 3 values? Or would one for each corner be better? Or even 5 like one in the middle?


How do you imagine such a scheme would work?


  Also is the capacitance linear with distance? Is there any formula?


If you mean with distance along the strip from the connection point then no.


  I have lots of copper tape left from guitar shielding, but it corrodes over time. Will the capacitance change with it corrosion, rendering it useless?


The copper oxide layer should be quite thin and should not affect the conductivity of the tape significantly. You can spray it with a protective coating to prevent this. A PCB conformal coating may be suitable as it is designed to protect copper.