I am about to replace a 1950 gravity gas heater with a modern unit, add c/a and do a little basement duct work to a unit all for $5,800.Around here (Delaware County, PA - near Phila), a "3Br, 1Ba, unfinished basement and no c/a" is the "base unit" for the row homes I buy. 

If they will be using your existing duct work it should not cost you that much but like I said I am not familiar with prices in Texas. 

I would say, though, if you have a furnace and duct work already, central air should cost more like $3-4K.

If you already have forced air heat just do what you've been doing and use the existing duct work.

. - they did plumbing, electrical, demo interior, installed duct work, new sheetrock and sub floor,  kitchen and bath cabinet, counter, sinks, tubs, toilets, tile work, and new windows throughout.- additionally, he purchased light fixures- interior doors- and appliance packages for each unit which are all sitting in storage now.- since he did not do the roof, there is now water damage from the leaky roof, to some of the new sheetrock and sub floor.Here is where we have gotten so far:**read a bunch of info on NOI, Pro forma, CAP, and looked at other delapated multi family buildings that have sold in the area. 

Watch out for duct work attached they may posed hidden costs also.

Not sure about the trim, but this looks like a basement - looks like ductwork on the ceiling.

Should I add ductwork and HVAC or add mini split “ductless” systems?

Does this mean I have to facilitate heat by providing the furnace, ductwork, etc (which I obviously would) or does this mean that I actually have to continue to pay for the fuel?

You mentioned the lint trap, that should be easy to check, but also add a secondary clean out in case the problem is in the duct work.