Are you talking of the cockpit floor or the hull? I don't think that any pumps are on the cockpit floor (which is balsa cored also). The plastic hatches are usuall silicone sealed--although Life Caulk, life seal is probably a little better, since it is specifically for plastic and has both urethane and silicone.
The best way of taking care of the screws in the cored hull is to remove them, and if possiable fill the hole with epoxy--see my piece written today on drilling a hole, and under cutting.
It depends on what the base of the pump is made of --unfortunately some of the pumps used on the c dories are plastic and either of the pure urethanes is not recommened--see above. On my 25, the main bilge pump is screwed into solid glass in the sump in the center of the hull. under the inspection plate--I am not sure how the 2006 boats are constructed. I also have a macerator pump--which has rubber mounts as well as ss "legs". This was into the core--I pulled it and put in the plugs as above.
Unfortunately bilge water seems to sit in the bottom of the C Dory25 more than some of the other boats.
Yes, the hull must be dry, as must be any holes--including any core. If you use a urethane, use 4200 fast set. 5200 is too perment for a pump etc. I use 5200 for things which are not going to come off.