So "det finns inte kaffe" would mean "there isn't coffee" and "det finns inget kaffe" would mean "there is no coffee," with special emphasis on the noun? Can you even negate "det finns" with "inte?"
Also — because this is "inget" and not "ingen," does that imply that there is no coffee, say, anywhere in the house, as opposed to there not being a cup of coffee?
Yes, det finns inte kaffe would be there isn't coffee, and det finns inget kaffe means there is no coffee, I don't think there needs to be any special emphasis though? It's just that inget/no refers to the noun.
About ingen/inget, yes, and therefore it will almost always be Det finns inget kaffe, because it is a bit strange to speak about en kaffe that way unless there is some very specific context.