It is ruled a dead ball and each base runner would advance one base.
no dead ball
The catcher needs to catch the ball because the ball is still a "live" ball. (Unless it's a foul ball that the catcher wouldn't likely catch that anyway. Any base runners may advance when the ball is "live" at their peril. If the ball is not caught, there isn't much peril to runners running to the next base.
If the pitched ball four (issuing a walk) gets past the catcher, it is still a "live" ball, and the batter, and any other runners on base, may advance as far as they can. The catcher, or any other defensive player must retrieve the ball to keep it in play and prevent the batter or baserunners from advancing.
Yes, if the ball is pitched in a location where the catcher has a reasonable chance of stopping the ball, that is a passed ball and the error is charged to the catcher. If the ball is pitched in a location where the catcher does NOT have a reasonable chance of stopping the ball, that is a wild pitch and the error is charged to the pitcher. Actually, a passed ball is NOT charged as an error against the catcher. It's simply charged as a passed ball. Not terribly logical, I agree, but that's the rule.
The vast majority of Balks are called on the pitcher, but a Balk can also be called on the catcher. With runners on base, if the catcher steps out of the catcher's box before the pitcher releases the ball, it's Balk on the catcher. When this happens, it is almost always while attempting to give an intentional base on balls.
Ball's dead and runners can't advance.
In general, if there are base runners, the pitcher should run to cover home. If there are no runners on, the pitcher can just stay on the mound.
The following is copied from MLB rule 5.09(g): " (g) A pitched ball lodges in the umpire's or catcher's mask or paraphernalia, and remains out of play, runners advance one base;Rule 5.09(g) Comment: If a foul tip hits the umpire and is caught by a fielder on the rebound, the ball is "dead" and the batsman cannot be called out. The same shall apply where such foul tip lodges in the umpire's mask or other paraphernalia.If a third strike (not a foul tip) passes the catcher and hits an umpire, the ball is in play. If such ball rebounds and is caught by a fielder before it touches the ground, the batsman is not out on such a catch, but the ball remains in play and the batsman may be retired at first base, or touched with the ball for the out.If a pitched ball lodges in the umpire's or catcher's mask or paraphernalia, and remains out of play, on the third strike or fourth ball, then the batter is entitled to first base and all runners advance one base. If the count on the batter is less than three balls, runners advance one base. "
An umpire is not supposed to throw the ball back to the pitcher with runners on base. If he does the ball is dead from the time he touches it to the time the pitcher has the ball on the pitching mound. So the runners cannot advance.
If it was a hit ball and the ball was stuck in the runners jersey then the runner is out assuming he had the first contact with the ball. If a fielder has the ball stuck in the jersey then the ball is dead and the hitter would be safe at first.
In a normal Base On Balls call, the catcher wouldn't have to throw the ball. If it's a passed ball or a wild pitch, the batter is still awarded first base and can't do anything else. If there are other runners and they try to advance and the batter interferes after that, he is certainly called out and the runner will have to go back to the base he started from. But on a normal base on balls, the ball is dead and the runners would only advance if they were forced and the catcher wouldn't have to throw the ball anywhere except to the pitcher.
In the event of a batted ball being hit on the fly and was caught by an Infielder or Outfielder on the fly before it hit the ground, it will be ruled as a Fly Out and in the event of there being no outs, one out or two outs and in the event of there being runners on base, runners will have to return to their original bases prior to the pitched ball before they can advance to the next base.