I have a 98 Intrepid with a 2.7 engine. It was driving fine on Thursday night. Went to leave for work Friday morning and it wouldn't start. So far, we have tried a new battery, had the started tested,

Check that timing belt has not broken. Happens frequently at cold start.

Has the battery been load tested? do that first.

crank sencer

This is all tied to the service engine soon light.

When the Check Engine light (or Service Engine Soon or small picture of an engine) is turned on, the systems computer control unit (called the PCM) has sensed a fault in the electronic controls of the engine and/or the emission system. When this happens you will seldom feel the car engine run any different. There are about 200 possible system faults (the gas cap is just one of them) that turn the Check Engine light on, to know what has developed a problem you must have the system tested, this is called an OBD2 fault code test, Kragen/Oriley and Autozone parts stores (USA only) do this testing for free.

Check for sparks.
Do It Yourself Diagnosis and Repair

Start by checking your battery cables for a good clean contact at both ends. Also check main fuses.

You need to find the source of the leak. it hasn't overheated yet because you keep catching it and re-filling. From your description, it sounds a lot like a bad water pump seal (part of pump) but without physically looking you can't be sure of that. when the pump seal begins to fail, internal pressure while running often will drive the seal tighter against the shaft stopping the leak. as pressure drops as the engine cools, the seal relaxes and permits seepage.
With time, that will change and it will leak all the time.

could be a air pocket in your lines

try spraying the linkage on the side of the carburator with some kind of lubricate W-D 40 works well or check the strenghth of your spring that