Is it the fuel pump or distributor cap?
I have read your question three times and it sounds like there is more of a problem with the starter than the fuel pump etc. When the temperature drops the resistance in the starter increases, if there is a problem with the connections between the battery and the starter it will be aggravated. If I was you I would remove the battery connections, clean them with a wire brush, refit and spread petroleum jelly over them to prevent rust build up. Do the same with the connections on the starter motor. This will hopefully solve your problem.
