Water – this pipe in the basement

High efficiency furnaces will make condensate. They run the air back through the hot exhaust to get more heat out. This also cools your exhaust enough to use PVC - that's what your upper line is.

In the process of cooling the exhaust, you might get some condensation. Typically that's collected in a little sump tank. When the sump reaches a certain level, a little bitty pump pumps it out - that's your lower line.

If it freezes where you are, you might want to insulate that lower line. If it freezes shut, your sump could overflow in the mean time. It's just a dribble of water, but still. It's a leak of water in your attic.

Yes, that's definitely black mold. Mold doesn't take a lot of water to grow, but from the looks of it that one is still actively growing.

The brownish hose in the picture looks like clear vinyl tubing. If that's a drain extension, where does the main hose end, right at the wall flashing? Could the joint between these two be leaking into the wall? Or if the brownish hose comes loose, could the main hose drip/drain into the wall? Like Ed Beal said, an a/c unit can condense several gallons of water per day, more than enough to cause the mold you see if it's leaking into your wall.

As Lee Sam mentioned, I noticed the gap at the top of the box where the hoses come through the wall. Any rain that hits the siding could run down, land on top of that box, and run into the wall. I didn't notice any streaks on the siding to indicate such drippage, but there is rust on the top edge of the metal access hatch beside the heat pump, under the siding.

Whether the water comes from the "seepage" issue, rain leaking in on top of the vinyl box, or the condensate drain, the leakage fortunately looks very localized. Unfortunately, there could be further damage inside the wall including more mold and wood rot. You'll have to remove the fiberglass in that corner above the foundation and that piece of drywall on the basement wall to see. If you may pursue legal action against your neighbor or the condo association you'll need good photos and documentation of anything you or a professional does.