I believe the unit has to be a heat pump (AC/heater) with an in unit condenser. 

This property is vacant and winterized, but there is so much moisture in the basement, the water is actively condensing and dripping off the fluorescent light tubes.

You've probably already nailed most if not all of these calculations in your business classes, but doesn't hurt to have them all condensed in a single reference book.

Plus, no condensation on the cold line.

Got to see inside the sump and it seems kinda - what we used to call - groty.Now, it's not an effluent sump, it only serves the floor drain next to the furnace for AC condensate and humidifier waste water.

Vacant houses are subject to getting the Air Conditioner Condenser ripped off.  

The forced air furnace were installed I just never realized there were no condensers on the property, which would have been the first sign it wasn't included.

The condensation that happened because of the large area of the plastic mat makes perfect sense. 

A new condensing boiler may be your best option.Mini splits are cheap enough and easy to install.

Although it was 3 years old,, So life of a window A/C is maybe 5 years, so it's up to you, but realize it was close to 50% wear and tear life left, the weep hole was probably clogged and that's why the guy drilled a hole in it because condensation was probably building up and spitting or dripping inside.