Is it callled the Batcave because it is a bat inhabited bat cave or because it is batman's possession like the Batmobile, Batcopter, Batplane, Batboat, bat etc., etc.?

If there aren't any bats living in Batman's cave in a particular version there is no bat guano to worry about in that version. Or possibly there is a dividing wall between Batman's stuff and the bat habitat.

Possibly the Batcave is climate controlled with heat, air conditioning, dust filters and dehumidifiers and humidifiers as needed. Thus the equipment would be just fine in there.

Is it called the Batcave because it is a bat inhabited bat cave or because it is Batman's possession like the Batmobile, Batcopter, Batplane, Batboat, bat etc., etc.?

If there aren't any bats living in Batman's cave in a particular version there is no bat guano to worry about in that version. Or possibly there is a dividing wall between Batman's stuff and the bat habitat.

Possibly the Batcave is climate controlled with heat, air conditioning, dust filters and dehumidifiers and humidifiers as needed. Thus the equipment would be just fine in there.