Filtration soiling came up in home inspection

[Shane Smith]

My wife and myself have found our dream house but are concerned about an unknown dirt or soot that is in the home. Home description: ranch style home with basement in eastern NC, no crawl space, central heating and air. In the bedrooms along the base board is a dark dirt tint on the carpet, roughly 1/2" all around the room (evenly). This is in 2 rooms that are located above an unfinished basement. We have been told the carpet was cleaned 2 times and they were unable to get it up.

Separate issue that may be related but unknown, in the living room and kitchen a slight black soot is collecting on some of the windows, ceiling, and oddly enough inside the refrigerator. It wipes away relatively easy with a finger. In the living room there is a gas log propane system that is completely clean with no black soot on it. The previous tenant had multiple propane tanks at the house after moving out. Are these oddities connected or two separate issues? Online research points to 'filtration soiling' and it seems to be somewhat common down here (NC). We're ordering a mold and moisture test, if they come back good (pass) should we be concerned? Should we walk? Any thoughts?

My uneducated synopsis: House has an issue with cycling air throughout the house properly (pretty much filtration soiling caused from improper placement of insulation). The tenant that lived in the house previously left a bunch of propane tanks behind which seems like they relied on running the gas logs to heat the house. If the house wasn't properly cycling air then the black soot was getting drawn around the house. 

[Mike McCarthy]

Is propane the only heat source? I ran into something similar with a flip that had an oil burner, and the oil soot was circulating in the house due to a bad burner.

I didn't think propane could cause similar soot (but I'm far from a propane expert).

Other than the carpets, does the spot wash off with a cleaning solution?