It still is getting 3bars when the pump is priming, so that is not what is preventing the car to start.
I really think you dont have ignition there (you DO have a problem in your fueling system, but it should not prevent your engine from starting).
I dont know the 2jz much though so cant say why you dont get spark, but this is a generic issue. Are you sure the timing belt is set correctly ? Is it possible to set it wrong like with the crank having a turn or a half turn more than it should ? (cant explain it in english, sorry, maybe someone can explain what i am trying to
) That would explain the pop / bangs too .
Keeping the foot down to start mostly means the car is overfueling, so that could be very leaky injectors ... are they the stock ones ? If not which ones are they (please dont say venom / TLF /TRE performance, if so you found the cultprits)