The fuel pump doesn't have any effect on your engine bogging down before getting up to normal RPMs during start (unless it's completely gone TU and you crank and crank until you kill your battery). An inop fuel pump means your X won't run. It's possible that you have a fuel leak somewhere, but my bet is your battery is toast. Cold weather will weaken the battery, especially if your electrolyte level is low. Weak battery and / or loose cables, which are very common problems on the Xterra, will make it hard to start, and even harder in cold weather. The battery cable clamps (even on the second gen) are absolute junk.
Go confidently in the direction of your dreams! Live the life you've imagined. As you simplify your life, the laws of the universe will be simpler. - Thoreau