If you have a rod sweeping out an area dA in time dt, this is the same as the rod of length L moving a distance ds in time dt. Given that the rod moves perpendicular to the field, then the area swept out is L ds = dA. It's the same as picking up a piece of lead graphite and scraping it on the edge, you form a rectangle.
As for the bar magnet, emf = -dN(phi)/dt ; the emf induced is proportional to the rate of change of flux linkage through the coils. Flux density, B, is the flux per unit area, so flux is just the quantity of itself. Flux linkage is the total amount of flux through one loop multiplied by the number of loops that flux passes through; N(phi). At A-Level you assume that the flux 'fills' all space within the loops. Obviously, if you think of the extreme such as a 20 mile radius loop with a small bar magnet in the middle, the flux is not going to fill the whole area of that loop.
But for sufficiently small loops, it is safe to assume that the flux from the magnet fills the whole area of the loop/s.