a. Bodies moving under a gravitational force (or any central force) follow trajectories that are conic sections, i.e. hyperbola, parabola, ellipse or circle. Depending on the energy of the body it has on of those orbit. Most physics textbooks have a description, under Kepler's Laws, Central Forces, gravitational force or whatever. Look at Particle Mechanics by Collinson and Roper, or Classical Mechanics by Kleppner and Kolenkow.
d. If you are unsure, why not simply re-derive the formula for the potential energy from U(x)=a/x^2 - b/x ?
What do you mean "get rid of the x's"? You want velocity v as a function of x don't you? So re-arrange
0.5mv(x)^2 +a/x^2 – b/x = -b^2/9a
to get v(x) = ...whatever...
The rest looks ok to me.
e. Draw the potential energy function U(x). Check where the initial condition is on that graph. The particle can only stay in region at the same height, or lower, that that of its initial potential energy. That is, given the initial condition xo for example, the initial potential energy is U(xo), and the particle must always have a potential energy equal to or lower than that. So you can draw a line through (xo, U(xo)) parallel to the x-axis, and the particle is only allowed to move in that region between this straight line and the graph of U(x), by conservation of energy. Then you can find the maximum and minimum x's; it's where the straight line cuts U(x) again, or if it doesn't, +/- infinity.