In order to find the PDF for Y = X^2 you need to use something like the transformation theorem of the characteristic equation. Do you have a particular PDF in mind? Here is the transformation theorem (scroll down to change of variables formula):
Transformations of Variables
With regards to your other question, look at the definitions for expectation and marginal densities. The marginal density is used to "integrate out" the other variable so that you get a PDF in terms of one variable.
Also remember that E[XY] = Integral over region xy*f(x,y)dxdy.