I'm guessing that the paper has been supported by another device (table, pad, etc.), correct? Try writing on a piece of paper being supported by nothing but the UA.
What, you mean like layers upon layers of rock?
I don't think you quite understand the ratio of finite to infinite.
No matter how thick it is, it is infinitely wide, therefore has a smaller thickness:length ratio than paper.
Doesn't matter how many layers of rock there are.