2002 Mitsubishi Eclipse V6 Convertible

As it says in the title, the engine will turn over but will not catch (mostly).

I have been experiencing leaking on my cloth top for a few months and usually cover the car when a heavy rain comes, but had one crop up on me without me realizing and the car got very pretty wet inside. This might be important because water pooled under the foam back seat and I recently discovered that the fuel pump is installed directly under the middle of the back seat area. I have tried turning the key to the ignition setting and don't hear anything from the fuel pump, but that may or may not mean anything.

After the heavy rain, on a very cold morning my car failed to start so that I could get to work. I just replaced the starter a few months ago so it turns over with no issues as long as the battery is charged. But that morning, and most every time since, when I try the engine will not actually catch to start. 

Further backstory, a few months ago (summer) a radiator hose sprung a leak and the engine ran hot for two weeks with me fighting it with coolant while I tried to locate the leak. There does not appear to be any coolant in the oil, but the engine is fairly old and probably has a couple of exterior oil leaks that aren't easy to find due to grease and grime.

I have pulled one of the plug wires and put in a spare plug to confirm that there does seem to be a spark getting to the engine. I do not have the tools to perform a compression test at home and the only code that is throws is an EGR recirculation valve one that it has had since the hose incident but I have also been unable to pinpoint as it does not appear to be the valve itself (likely the solenoid). 

Finally, I say mostly unable to start because while looking at it again today it actually started to act like it was going to catch and then did. It ran (a bit roughly) for about two minutes before dying. While it was running there was a bit of white smoke coming from the exhaust. 

I was pretty convinced that it was a fuel pump issue. Especially when I pulled the backseat today and noticed how damp the foam of the seat was and how close it was to the wiring for the pump, but after removing the protective cover it seems to be plenty dry in there. Also, when it was running for the few minutes it was responding to the throttle just fine which doesn't seem like something that would happen if the fuel pump was the issue, but I am just an IT/Computer guy who dabbles in auto repair by necessity more than anything else.

So, does anyone have any further suggestions or ideas, cause I'm having trouble narrowing down where else to look for possible root causes?

[Update] The issue was the Camshaft Position Sensor, which is part of the distributor assembly on my vehicle, so I replaced the entire thing and solved the issue that I was experiencing once I finished installing the new assembly and recharging the battery.