2001 Ford ZX2 motor transmission strain, now leaking antifreeze?

At first we wondered why it was starting to sound like it wasn't changing gears properly. Yesterday, I saw that under the front center of the car, it had a very fast drip of antifreeze. As far as I know, the water pump has been replaced twice over the past 2-3 years. Unless it is running on top of a guard and going over to the middle, what else could it be? Yesterday afternoon it had a heavy drip. As of last night, it magically stopped and the overflow tank that we thought would be empty was still as full as when I filled it up yesterday afternoon. What gives?

Added (1). Transmission was rebuilt just about a year ago with no issues. Starting to wonder if that isn't related to what is going on now. Transmission fluid was serviced at that time and is full and clean.

You might have 2 unrelated problems. Without seeing the leak it's hard to say for sure where it's coming from, but a common place for Ford Zetec engines such as yours to leak coolant is from the thermostat housing. The housing is typically a plastic piece and it tends to crack inside where the gasket inside the housing seals to the engine. The housing is located on the right side of the engine as you look into the engine bay. Look underneath or around that housing for a leak. The leak can be intermittent at first but it will become constant as the crack gets bigger. Replacing the housing fixes the problem.

As for the rough shifting, it could be a result of a bad mount (or it could be a genuine transmission problem). It's easy to check the hydraulic motor mount - it is covered by a black plastic circle and is located on the left side of the engine as you look into the engine bay. You can remove that plastic cover to inspect the mount. The mount should look like a fully inflated balloon. If it looks like it's deflated, then the mount needs to be replaced. If it's deflated looking, the engine might also be slightly tilted (lower on the side where the mount is located) as it sits in the engine bay. A bad hydraulic motor mount can cause the engine to "move around" a little during gear changes and cause a significant thunk/vibration/shudder that can feel like a transmission problem. If the hydraulic mount is okay, it could still be the transmission mount and not a transmission problem but at that point it might be best to let a mechanic take a look.

Just offering some suggestions based on past experience working on Ford Zetec powered cars.

Quit messing around guessing. If the transmission cooler is getting coolant into the transmission you'll be walking soon! Water pumps leak near the drive belts, not in the center of the vehicle. Look at the coolant level in the radiator not just the reservoir. If that's the 4 cylinder motor I've seen problems with the plastic thermostat housing deteriorate and leak. You can't see where the leak is because it happens on the bottom of the housing but it will drip down the area between the motor and tranny. I'll bet that's your problem. Probably a $350 repair.