Re: String Theory (M Theory)
Re the original questions:
No and no.
You can presumably draw an n dimensional space on an n-1 dimensional surface as long as some convention regarding the extra dimension is made, relating it to an "angle" on the n-1 dimensional surface. If you try to add further dimensions the problem it will no longer be single valued (I think they call it the property of injectivity in mathematics) making the diagram useless. If you want to draw a 10 dimensional shape you would need a 9 dimensional sheet of paper.
Mr P. I think you are correct in saying that string theory is not presently experimentally testable, and is therefore not a physical theory.
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