Visited Biltmore several years ago and did not see or hear of any outdoor pools. That’s not to say there might have been one on the (extensive) grounds at one time.
Correct. There are several artificial fish and special vegetation pools, but no "swimming" pools. There are also a couple of artificial lakes. I doubt there was ever any other swimming pool except the one in the basement which was part of the gym/showers/bowling alley arrangement.