The Dodge Neons suffer from water leaks around the rear tail light areas caused by the foam seal and/or trunk seal weatherstripping leaks. Solution is to pull the light out, apply silicone seal to the foam seals and put them back on the car. The trunk seal can be done in similar fashion. This is a known issue.
Heater Core is the likely culprit
this could be from the heater core. When the heater core goes out it leaks....right into the passenger side floor board If it is water, not coolant, the problem could be a plugged air conditioner drain. Easily accesible from under the hood.
Check your heater core under your dash.
evaporator drain tube plugged
Yes, it can.
Water is leaking on passenger side floor board when heat is on
If you put it on jack stands, yes.
Failed radiator transmission cooler.
Spilled water can make a floor slippery, which can cause people to slip.
Water in the fuel will keep it from starting.
You should immediately try to mop up, or remove as much, if not all, of the water as you can. Leaving the water on the wood floor will cause it to warp out of shape.
check battery if it has water and check the starter