Disconnect the remote starter and Retest. You may have to remove the lower knee bolster to gain access to the ignition wire. The Black/White wire is probably spliced. Disconnect any splice connections you see and connect them back up to stock.
A lot of times the remote starters will interfere with the starting signal. If it STILL doesn't start, make sure you have voltage on the large connector on the Drivers side Strut tower. Should have 2 or 3 large wires in this connector. Have a friend turn the key while your testing for voltage and confirm you have 12v+. Next step is to diagnose the Neutral Safety Switch. Confirm the NSS is working properly by checking the Resistances on certain terminals and confirm its outputting the start signal to the Starter itself. The starter has two wires going to it. A Small Clip style wire and a Large Terminal style wire. The large wire goes directly from the battery, the small wire goes through the NSS To the starter. If the NSS is not detecting PARK or NEUTRAL the car will NOT start. Try putting the car into Neutral and starting it. If it starts right away, the NSS Is cuplrit and should be repaired.
Before tackling any of the wiring, you should do the easy things. Confirm the Relays are working properly under the hood. Confirm your fuses are not blown under the hood.
You've already confirmed the fuse panel under the coin tray was OK so that's not in Question.
Here are a few helpful links to help you out:
Neutral Safety Switch Cleaning:
DIY: Neutral Safety Switch Cleaning
HELP My car wont start!
DIY : HELP 92-96 5sfe CAMRY WONT START
Starter Repair:
DIY: 5sfe Starter Repair