I have not had a problem with regular trim but those darn baseboard radiators I will take recommendations on  bulletproof options there.

There will invariably be something on the home inspection, but hopefully it's not expensive, or something that you can live with.Personally, I find better value in fixing up a rental - my first rental needed more than a dozen new windows (some were in rough shape, and there were even taped storms) and I added central air (removed radiators).  

In the case hydronic in-floor/wall/ceiling radiation systems, solar thermal can provide sufficient heating on its own. in the case of hydronic baseboard systems, it can serve to preheat the return lines to lessen the load on the boiler.  

The higher the heat bills the lower your rents will have to be to compete with newer easy to heat apartments.Also when these same old buildings have the old hot water radiators tenants will complain about being too cold or hot unless they have their own thermostat.Shop around a lot before buying...

Some apartments will be hot like .... and others will cry.Most brooklyn apartment buildings are one pipe and each radiator have an air valve.

Is it being used to radiate heat from the sun into the apartments?

The current heating system is an oil fired steam boiler and radiators.

"Solutions-Bleed the air out of all the radiators: Bleeding the radiators is a necessary maintenance step, usually done at the beginning of each heating season.

Heating system  needs to be replaced, it used water radiators  with an oil furnace.

On the first floor, there are pipes going up the sides of each of the windows where a radiator is located to shoot water up to the second & third floors.