Hello All, I have an update.
I have finally had time to mess with my Mazda, and I may have solved the issue. I know I said this before, but I am like 69% sure that I have found the problem.
Over the past few weeks, I been unable to get my Mazda started. It would crank, start for a half second sometimes, and die. I wasn't sure if my timing belt broke, due to it being able to kind of start, but I decided to check it anyways. I took the valve cover off, only to find the timing belt looking fine to me.
But, when moving the wire harness that goes over the valve cover, I followed the connector and wire Crankshaft Position Sensor down to the sensor and notice an odd route that it traveled. It seem like it was extremely close to a pulley.
I had another sensor to put on the P5, so I decided just to do it sitting at my house, instead of trying to get the car to my garage which is 25 minutes away. I got the sensor off, and I found that the pulley was rubbing on the wire, and shaved about half way through at the deepest point.
I installed the new sensor, routed the wire differently, and put everything back together. I reset the ECU, and the car instantly started, and ran flawlessly.
This was not the first time that I reset the ECU and the car ran better, so I took it for a few test drives, and it ran great, not one moment of hesitation, stall, nor surging. I am not convinced that this is definitely the solution, and I probably won't be until next fall when the temperature drops again, but the P5 was running really well today.
I am attaching a picture of the wire, I am not sure why the wire was routed the way it was, it actually looks like it is supposed to be routed that way, but if there is slack in the wire, it will rub on the pulley.