1) To draw indifference curves.
Assume some value of U on the LHS, assume some values of x and calculate corresponding values of y. for e.g. assume U=10.
x=0,0.01, 0.04, 0.09, 0.16, 0.25, 0.36, 0.49, 0.64, 0.81, 1 you get
y=2, 1.8, 1.6 , 1.4, 1.2, 1, 0.8, 0.6, 0.4, 0.2, 0.
Join these (x,y) coordinates to get indifference curve for U=10. Repeat for other values of U. What does this curve mean? It means any (x,y) on this curve gives you the same utility U=10. Therefore you are indifferent between all points (x,y) on this curve.
What you did is right mathematically.
The MRS is the slope of the indifference curve. It is the rate at which you must substitute y for x to remain on the same utility curve. i.e. suppose you are currently consuming a bundle (1,0) . You now want to consume some y, how much x must you substitute out to get to consume 1 unit of y at this point? That's the question the MRS answers.