But then the air from the bathroom would be sucked out, instead of the air from the outside being sucked in. This would expose the entire building to odors and bacteria from the bathroom, rather than containing it within the bathroom and having it processed through normal air filtration, not to mention the risk of the door opening on the face of someone passing by or getting ready to enter the bathroom.
No, this is fraught, extremely fraught. Leave well enough alone.
Not really. Air flow direction is a result of pressure differential, created by air in vs. air out.
The door swing direction is essentially irrelevant.