Part a is just asking for the area between the two curves. Basic subtraction. Take the area of the upper curve and subtract out the area of the lower curve. Your intervals will simply be the points where √x=e^-x.
For part b, yes, you can use shell method. Your radius will be your distance from the x-axis (which will be y), and your height will be the distance between the two points on either curve (again, basic subtraction. One function - the other = distance between).
Part c I can kind of conceptualize, but i'm failing to see how the region R is in tune with each cross-section being a semi-circle. Or how that semi-circle has a diameter attributed to it. But then, i've never been good with word-games.