Aftermarket cam and crank position sensors for these cars have been shown to be problematic. No guarantees, but often when aftermarket ones are replaced with OEM ones, problems like this go away.
(I can tell you that the aftermarket O2 sensors for our cars are absolutely incompatible - they have a major and documented electrical design flaw
, but for some reason, the manufacturers are either ignoring the problem or simply aren't paying attention and continue to sell O2 sensors that are seriously problematic - the exceptions being NTK/NGK and Denso. OEM, NGK, or Denso only for O2 sensors for these cars. Based on repeated bad experience with aftermarket cam and crank position sensors over many years by many people on the LH car forums, it is suspected that they don't necessarily have a design flaw like the aftermarket O2 sensors, but have poor electrical tolerances and reliability/quality issues - so OEM only.)
...THe ASD relay does energize when you turn the key on for one second but doesn't come on for about five seconds when starting it...
When starting, the PCM keeps the ASD relay off until
it is satisfied that it is getting good engine sensor signals, then it energizes it, and like magic, the engine starts. This would explain the delayed start, and, again, points to a marginal sensor issue. If it were me, I would replace the sensors with OEM and see where you are. YMMV.
Again - fuel pump or fuel filter issues may be at play. I can't be sure which (sensors or fuel pump) is the major player with your problems.