Question: Batter is hit by pitch on his hand. It's a foul ball, right, because the hands are a part of the bat?
Answer: No. If the hands were a part of the bat, it would be pretty hard to tie your shoes. If a player is hit by a ptich, he/she is awarded first base unless the batter has attempted to hit the pitch, the pitch was in the strike zone when it hit the batter, or the batter made no attempt to get out of the way of the pitch. It's the umpire's call.
Question: The base runner did not slide. They are automatically out, right?
Answer: Nope. The rule states that the runner must avoid contact. the rule does not require runners to slide. Sometimes sliding is the best way to avoid contact, but not always.
Now if a runner slides into a base and knocks over the fielder, the runner is not out. It sounds more like the fielder may have been covering the base improperly. However, it's the umpire's judgement whether or not the runner may have intentionally tried to take out a fielder.
Question: A pitched ball hit the ground, and then the batter. Dead ball?
Answer: No. The batter is awarded first base, unless he/she swung at the pitch.
Question: A pitched ball hit the ground and the batter swung and hit it. Dead ball?
Answer: No. Live ball. The result of the play stands (hit, out, foul, etc.).
Question: These new pitch count rules stink. I know my son/daughter better than anyone and he/she can pitch more. If I sign a waiver or something, can they be allowed to pitch all of the time?
Answer: No. And you're in the wrong league.