Some catchers are allowed more leeway to call pitches by their managers than others. But usually pitches are called by the manager or pitching coach and relayed to the pitcher by the catcher.
There's no minimum height for any position. Many catchers are among the shortest players on their teams, since height isn't much of an advantage for a player who spends most of his time on defense...
Mostly the catcher, although the manager has some say, and of course the pitcher is the one who actually has the final say in which pitch is ultimately thrown.
Cincinnati Red's Aroldis Chapman threw a pitch at a blistering 105 mph during the 2010-2011 season when they went to the playoffs and fell short in the first round to the Phillies. Still Aroldis is...
Well there are actually a couple really good catchers. The best I would say is Victor Martinez of the Boston Redsox with his great power, but he does't have the greatest arm in baseball. Then there...