1)If the car is carbureted then, yes you have a mechanical pump unless the TSB was performed back in the day, just look for the pump on the front of the engine block.
2)The mechanical fuel pump should be located on the right side (passenger) front of the engine block between the water pump housing and distributor.
3)Not getting fuel issues/problems can be related to a faulty fuel cap, clogged pickup tube sock in the fuel tank, clogged/restricted fuel filter, collapsed rubber fuel line or a faulty fuel pump.
a)I would recommend removing the fuel cap and see if the condition changes, the fuel in the tank expands as it gets hot, the cap has to be able to relieve this pressure (EVAP Systems) as the fuel cools down it contracts, the cap has to open to allow air to enter the tank or it will create a vacuum in the tank and preventing fuel from flowing.
I had this exact problem on my 72 Duster at the track, the car kept slowing down each round and it was a fuel cap issue not allowing air into the tank.
4)Testing a mechanical pump is quite easy.
a)You will need to remove the fuel inlet and outlet lines from the pump, if the lines thread into the pump I recommend using line wrenches so you do not round the nuts, and a brass fitting with a barbed end to thread into the fuel pump to attach the gauge, if you have rubber lines at the pump just remove the rubber fuel lines and attach a vacuum/pressure gauge and crank the engine.
b)SAE Line/Flare Wrenches
Amazing deals on this 5Pc Sae Double-End Flare Nut Wrench Set at Harbor Freight. Quality tools & low prices.
www.harborfreight.com
c)Vacuum/Pressure Gauge Test Kit
Amazing deals on this Fuel Pump /Vacuum Tester at Harbor Freight. Quality tools & low prices.
www.harborfreight.com
d)On the inlet side, the gauge should read vacuum and hold, on the outlet it should read pressure, generally in the 4-6 PSI range on carbureted cars.
Fuel Pump