As far as keeping feet dry...
There is a difference between a cold downpour in damp conditions and downpours in other conditions. For situations where it isn't cold, I don't worry too much about it and just let my feet get wet and then dry again. When it is cold, I've used shoe covers though have difficulty finding them in my shoe size so have also improvised with plastic bags inside the shoe.
As far as slipping from the pedals including wet:
My bikes still have a few different attachment mechanisms. My Trek 520 still has toe clips so that also addresses the slipping issue. My Trek 4500 has two sided pedals - platform on one side and clipless on the other. Most of the time, when commuting or around town, I use the platform side. They can be more slippery when wet but it hasn't been a big issue.