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.
Thanks to Femi Ogunsola for suggesting that I switch to z as a dummy variable of integration rather than using r which was specified to be the distance of the center of the loop from the wire.