x2 The prefab would also make an excellent brooding house which if placed next to the coop and run you are custom building, with chicken wire between the two spaces, will allow for constant 'Look, Don't Touch' of your chicks for super-easy integrations, or if using as a chicken hospital, will allow the sick bird to better maintain their status in the pecking order.Build the coop you want, and use the one you have in the meantime (assuming there's room). It'll buy you some extra time and then you can reuse the prefab as an isolation set up.
As far as covering the run, obviously if you build the one you want that'll be covered. Otherwise the chickens can manage rain by themselves and covering the run would mostly be for predator protection and shade.