vertically mounted fan with rain cover--is it a good idea?

Another option is a Turbo Maxx.

I have one. It mounts to the existing roof hatch and rests on a foam gasket between the roof and housing. The hatch can be left open and the fan thermostat set for auto run and shut off. Mine is over the bathroom and rain has never gotten even during severe thunderstorms. I do close it during tropical storms and near miss hurricanes.