Yes, once instance when the batter would give a signal is when (s)he understood the signal that was given by the third base coach. This signal could be the batter touching the helmet after the signal was given by the coach or maybe reaching down to grab a handful of dirt. Sometimes you will see the batter call time and walk down to the third base coach to have a discussion. This could be because the batter did not understand the signal given by the coach.
An MLB batter can be replaced anytime during the game but once the batter is batting the coach must wait till he is on base or after their team bats.
because one of his teammates will bring it to him.
A batter can bunt anytime ...whether it's wise to bunt is a different story.
A batter is out anytime a fielder catches a foul fly ball.
Anytime a batter is walked it does not count as an at bat
Only the players on the field,the umpire,the coaches,and the batter or the runners. Nobody in the dug-out can call time out or the fans watching the game. You may wonder why the players on the field or the runners/batter can call it is because if their shoe laces are untied and if they are hurt in anyway that the umpires and coaches need to know about if it's serious.
It could be signs that the coach is giving the batter or as a form of cheering or encouragement
No, but if the other team thinks it is going happening, the batter will probably get one in the ear.
Anytime a third strike is dropped the runner can advance to first unless he is tagged or the ball is thrown to first before he reaches.
No, as long as it is a fair ball. Once the umpire signals the infield fly rule the batter is automatically out. However, if the ball is dropped and is ruled a foul ball, the umpire reverses his call and the batter continues his turn at bat. Nevertheless, the batter can not reach first from that batted ball. You will often find an umpire state "Infield fly, Batter is out if Fair". When the rule is in effect, the batter may not get on first base.
Anytime a batter is at the plate, a pitcher and catcher can agree to throw nothing but balls to that batter. However, no pitcher would do so without getting an order from the manager to do so -- but, again, the manager can make that decision at any time the batter is at the plate. It is NORMALLY done as soon as the batter arrives at the plate. But, if a manager orders a pitcher to throw four un-hittable balls after the batter has swung and missed badly on the first two pitches, that would not be a violation of the rules.
Anytime the batter is put out at 1st base on a batted ball it is not considered a fielders choice, it is simply a put out. Now in your scenario if the catcher would have thrown home and the runner coming from 3rd is out and the batter is safe at 1st, then in that case it would be a fielders choice