a) B is not uniform over the area enclosed by the conducting loop; you must decide how to set up the integral to evaluate the flux. B depends only on how far away z the point of consideration is from the long straight wire; thus spanning the rectangular loop with rectangular "slivers" of dA = a dz is a good way to proceed.
b) Now that the flux is evaluated, find the time derivative. Keep in mind that r is time dependent so use the chain rule for differentiation and replace dr/dt with v when it appears.
The terminal speed will be reached when the acceleration becomes zero. This happens when the net force is zero; that is, when the force on the trailing wire segment balances the loop's weight.
a) Let x be the distance from the right end of the rails to the rod and find an expression for the magnetic flux through the area enclosed by the rod and rails. Set up the integral for the magnetic flux spanning the area of the loop with rectangular strips with dA = x dr located a distance r from the long straing wire. After finding the flux, determine the induced emf by differentiating with respect to t (only x depends on t!).
b - e) Remainder of solution follows my presentation in class on the subject of motional emf. In part d) however the magnetic field along the rod is not constant so you will need to integrate dF along the rod.