1987 Trans Am 305ci FI with 112,000 miles

Priority #1. Transmission(1,000 miles ago), MSD Distributor cap, MSD rotor, MSD wires, NGK plugs all have been replaced and car runs fine.Driving it from point A to point B i did not have a problem. When getting back into my car to start it, nothing happend. Full charge on battery, terminals were double checked and had no acid build up nor were they loose. Turn the key in crank position and again nothing happens. Battery gauge didnt drop, lights didnt dim, radio and all lights work. Pulled out the starter and with the amount of miles on the car i just went ahead and replaced it and still nothing. i do not have a chip in the key and i am stuck as to what it can be. when i do go to crank it over i do hear a relay clicking around a diaphram just to the upper left of the battery, which sits on drivers side of the car. Any help, thoughts or suggestions would be greatly appreciated.

#2. When the car actually starts up, on occasion it will just simply die out. Have to power brake it in order for it to stay on. After a while of running it this way it will simply stay on. It does it weather it is cold or hot and more so does it when it wants to. Again, any help, thoughts or suggestions would be greatly appreciated.