Answer to "does this car have an external fuel pump?"There is only one fuel pump in the 1999–2004 VW Golf, and it is in the fuel tank. The fuel pump and top of the fuel tank are accessible through a small hatch in the floor directly under the right side of the rear seat. The hatch is fastened with three screws to the floor. To get at it, remove the rear seat bottom. It's pretty easy. There are two fuel lines connected to the tank: one is a high-pressure feed from the pump output, and the other is a return from the pressure regulator on the engine's fuel rail. The likeliest place for a leak as you've described is on the high-pressure line between the pump output and the inlet of the fuel filter (item #22). The hose could be punctured, a connection fitting might be damaged, the filter might be clogged, etc. It's advisable to replace the fuel filter if you're under there to fix the leak anyway.
Answer to "problems starting in cold weather."Do you know what engine is in the car? That would be very useful information. The coolant temperature sensor is a common culprit for cold start problems. There could be other causes. If you can find someone who has a VCDS interface cable and a laptop with VCDS software (Ross-Tech), you can look into what's going on with the engine and make a more accurate diagnosis. It's better than buying and replacing parts based on guesswork.