Rule 8.01 Legal Pitching Delivery goes into detail about the only two legal pitching positions (the set position and the wind-up) but the short answer is yes, not only must the pitcher be touching the mound, his foot must be touching the pitcher's plate (or rubber) until the ball is delivered to the batter.
in ASA rules yes in USSSA softball rules no
No, you can not lead off the bases in slow-pitch softball. You can't even lead off the bases in fast-pitch softball.You have to wait until the ball passes the home plate.
Yes. The only time a manager can go to the mound and it not be considered a visit is if the manager believes his pitcher is injured. In that case the home plate umpire will accompany the manager to the mound and listen to the entire conversation to ensure that the manager and pitcher are not talking strategy but solely about how the pitcher is feeling.
yes they can be replaced at any time