Mold Growing on Top of Exterior Door

Mold goes away when the water goes away, so unless your toilet is leaking again, you have some other source of water or leak on the upper floor. This could be any number of things, even things not plumbing related, like water getting behind the siding, or a leaking roof with water running down the inside walls. As far as killing the mold that is there, I wouldn't waste my time. If you don't correct the water problem then the mold will return very shortly.

I need to figure out if there's still a leak or some kind of humidity problem. Is it safe to assume that because the problem got worse there is? Should I try to find the pipes and see if they're leaking?

Since you have had issues with the toilet in the past that would be the first area I would check. If the drain pipes for your upper floor are leaking then I would expect to see discoloration bleeding through the paint on the ceiling and the walls. If this turns up nothing then in the hole you have already cut, I would get a professional moisture detector device like the one a plumber would use.

These can help detect moisture and humidity in the air which can give you clues if you found the potential problem areas. Don't buy a cheap one because they don't work very well, I would instead see if a rental place will lend you a good one.

I think I'm seeing some black wood in there (need a better flashlight!).

This doesn't necessarily mean anything, it could be discoloration from previous water damage.

I would still like to hire someone to fix the plaster ceiling. Is it going to cost a huge amount extra if I cut through the mesh and they have to replace some of it? Does area make a massive difference in price for that kind of job? For what it's worth, I'm fine having drywall put up instead of plaster.

How big is the hole that you had cut? You can sometimes repair a hole by cutting a small piece of drywall and fitting that back in place, then using some thinset or spackle around the edges. Sand and repeat thinset or spackle two or three times until you have a smooth paintable surface.

IMO you are making several leaps here that don't fit. You've got mold below a window, spread out about 2 feet on either side, and you suspect condensation as the culprit.

You specify that the damage is old - years old by the look of it. Condensation happens every year, every season. Why would the damage be old, it should be fresh if it's condensation.

Finally - if condensation were the culprit, the condensation and thus the mold would be uniform. Water leakage through a window, however, is going to fan out below the window in both directions...

The answer is your window leaked at some point - years ago by the sound of it since the damage is old.

Solution?

Remove and replace ONLY the section of drywall that is damaged. Do not add a partial vapor barrier as that's about as useful as putting in an aquarium with only 2 sides. Mud and tape properly and repaint.