In Methods of Rationality, the Weasley twins refer to the Map as being part of the Hogwarts security system. So it probably gets the information from the wards.

Yes; remember that they speculate it was made by Salazar himself, and remember in ch43 Quirrel independently says:

Professor Quirrell sipped from his own waterglass again. "Well then, Mr. Potter, I shall freely tell you what I know or suspect. First, I believe the Chamber of Secrets is real, as is Slytherin's Monster. Miss Myrtle's death was not discovered until hours after her demise, even though the wards should have alerted the Headmaster instantly. Therefore her murder was performed either by Headmaster Dippet, which is unlikely, or by some entity which Salazar Slytherin keyed into his wards at a higher level than the Headmaster himself.

The twins never mention in canon seeing the Basilisk on the map.

(Salazar seems to have specialized in security - one of his other talents was being a Legilimens.)