Re: '97 Grand Marquis Cold Start
Your problems seem to be related to fuel pressure. Fuel related problems can often be traced to a dirty fuel tank. In tank debris can cause the primary fuel filter sock to become clogged. Additionally, initial cold start problems can also be caused by a leaking check valve. Check for pressure on your fuel rail after the car sits overnight. If the fuel pressure is low or non existent, then your check valve is probably leaking.
I had a bunch of fuel related problems similar to yours on a used '01 that was owned by an elderly lady living at the beach. In eight years, the car had only been driven 46,000 miles at low speeds, mostly around Balboa Peninsula, Newport Beach (top speed anywhere on Balboa is around 40 max). Although the car looked new and had been parked in a heated garage, the tank was full of "gunk" and the fuel filter sock was clogged. I don't know if the beach weather had anything to do with it, or what the lady put in the tank, but I replaced the fuel pump, in line filter (you do change your filter regularly, don't you?), and steam cleaned the tank. I've had no further problems for the last three years or approx 40,000 miles. A fuel gauge is a good diagnostic tool to have for these problems.