The fuel pump primes first thing, but also under certain other conditions which are time/temp related, it primes then stops, or it doesn't prime at all.
If it doesn't prime first thing in the morning something is wrong, a lack of cps signal can cause this, but so can the ecu, There is no harm in wiring it direct to the position 2 feed,(keeping it through the cutout).
The point about your non start is almost definitely the distributor cap, age old problem, inside the cap is nice and warm/hot, it is then blasted with colder water causing the moisture inside to condense, causing arcing/crossfiring/misfiring inside the cap, odd that it should do it the following day, but certainly worth a look, clean out with paper towel, spray with wd40 then dry out again, check inside ends of leads in cap also, but yes potentially could be water elsewhere.
The fact that the pump does not cut off in pos 2 with engine stopped suggest that either it has a sticky fuel pump relay, or the ecu is telling it to run, maybe the 4wd ecu is slightly different in this respect, maybe just tap the fuel pump relay or unplug then plug back in all the time whilst the ignition is in 2 without engine running, if tapping it stops the pump the relay is sticky, if it stops then restarts once realy is plugged back in then it is being fed by the ecu instructions to do so.
The relays are boggo std type a or type b relays nothing special at all, cheap enough if you need a new one.
tabetha