How to seal off exterior hole

Depending on your siding material, it is possible that the door isn't properly flashed, or the existing flashing needs replacement/repair. If this is the case, I would recommend two things

1) using some expanding foam, fill (but dont over fill!) the void between the door frame and the surrounding studs.

2) Properly flash around the door to make sure water can't get behind the door frame. Depending on the flashing material used, this may involve some caulk to seal the flashing (though peel and stick flashing won't require any caulk, and is quite easy to work with).

I am going to leave the info about the flashing, as it might be helpful to others, but in regards to your re-worded question-

Depending on where the water is coming in, you might need to either replace the rubber gasket/seal that fits to the outside of the door stop (if water is coming in on the sides/top of the door) or you may need to replace the door sweep (the rubber or bristle bit that hangs down from the bottom of the door itself, and seals everything shut when the door closes).

Primer dries fast, so 50's will probally be OK, especially if it gets some direct sunshine. Give it some time however, before painting. I have used exterior paints as low as the 40's as long as night time temps are not going much lower, or paint/primer at least 4 hours before temps drop a bit. Some paints do give a temperature range. There are low temperature paints that will cure down to upper 30's with no problem. The main problem with priming or painting in cooler temps, is that you must let it cure enough before applying the next coat. That may take a couple of moderate temp days as it will not cure in the colder nights, dew or excessive moisture, as that will delay the curing process or discolor the finish.