Idle relearn procedure helped someone with the exact same circumstances. However, they didn't get one good start out of the car. If the idle air control valve is not working well or is not controlled correctly, you might need to make your own idle with the gas pedal.
Fuel doesn't really have a pathway to the exhaust except through the exhaust valve if unburned. It would take a lot of fuel, and hot cats would burn that fuel. Water condensation in the exhaust might be fooling you with its drips.
You can confirm the fuel pump is humming away under the back seat for two seconds every time you switch the key to the run position. If it doesn't reliably run every time, the CCRM or one if its internal relays may be corroded. The PCM also gets its power through a CCRM relay. It may take up to 15 seconds of running (10 key cycles) to pump fresh gas from the tank through the fuel system, to saturate the filter, and back through the return line. Fuel pressure should be maintained at 30PSI at the rail diagnostic port for up to a minute after pumping.
If the car starts only about 1/3 of the time, but after repeated attempts you can get it to start, the camshaft position sensor may be on the fritz. That would cause no-run, not rough-run.
If you can get it running and leave it running for a while, some potential problems may clear themselves: sticky IAC, sticky injectors, fouled plugs, etc.
One part that could have gone bad that would make for poor running and make for actual fuel flooding the intake is the fuel pressure regulator. It uses engine vacuum to regulate pressure with a rubber diaphragm, and if this is broken internally, besides bad pressure, the engine can suck fuel through the regulator into the front of the intake. Apply a hand vacuum pump to its hose and see if it will hold 15PSI vacuum, or if instead you get fuel in the hose.