Situation #1 - Hitter has 2 strikes, swings at, and misses, a third strike that bounces off the ground and into the catchers glove.  Is this a dropped third strike? 

Correct ruling:  This IS a dropped third strike and the batter has a right to attempt to reach 1st safely.

Baseball-Rules.com explanation:
Dropped Third Strike
When a third strike is called, or is swung at and missed and the catcher does not make a legal catch, the batter may attempt to reach first base if it is unoccupied when there are less than 2 outs, or even when it is occupied when there are 2 outs. Occupied means it was occupied at the time of the pitch. The fact that the runner attempts to steal does not make the base unoccupied. Time of pitch is defined as the moment the pitcher starts his windup or commits to a pitch to the plate.
To be legally caught the ball must be caught in-flight. This means if the catcher catches the ball cleanly on a bounce it is NOT a legal catch. Rule 2.00 BALL casebook.
The batter may run to first anytime before he enters the dugout. He is not out simply because he walks off toward the dugout.
If the bases are loaded with 2 out and the catcher does not make a legal catch of a third strike, a force play goes into effect because the batter has now become a runner. The catcher may step on home plate to force out the runner from third or tag the batter or throw to any other base.
Rule 6.09(b) Official Baseball Rules

Situation #2 - Ball is batted or bunted in front of the plate.  Runner going to first is in fair territory (inside the baseline), not in the "running lane".  Catcher fields the ball and throws to first.  Ball is not caught at first.  Runner reaches safely.
Correct ruling:  This is an umpire's judgment call.  If he determines that by the runner being inside the base path, he interfered with the fielder's attempt to field the throw, the runner should be called out.  If he was hit by the throw, he should, automatically, be called out.

Baseball-Rules.com explanation:
Running Lane
The batter is out if, in running to first base, the batter-runner is hit by a throw while running outside of the 3 foot running lane, or interferes with the fielder taking the throw at first base. He could be called out even if he is not hit by the throw, if the umpire judges that by being outside the lane he interfered with the fielder’s attempt to field the throw. There must be a throw before interference can be called and the throw must be a quality throw. Rule 6.05(k).
A runner is not free from interference while in the lane, nor automatically guilty when out of the lane. If he is out of the lane he is in serious jeopardy of being called for interference, but it is not automatic, unless he is hit by the throw, or commits an intentional act of interference. The rule states that he is out when out of the lane AND causes interference with the fielder taking the throw. If he is out of the lane and is hit by the throw, that is always interference. If he is in the lane he could still cause interference, but it would have to be something obviously intentional (like grabbing the fielder's arm or glove, or deliberately touching a thrown ball). If the catcher does not make a throw because the runner is outside the lane, this is not interference. Interference with a thrown ball must be intentional. Such as, deliberately making contact with it. Or in this case if the runner is hit by the throw while outside the lane. The lines marking the lane are part of that "lane," but the runner must have both feet within the lane or on the lines marking the lane, to be judged as "in" the lane. Rule 7.09(k) casebook, N.A.P.B.L 4.14.
If the runner is hit by the throw or a collision occurs on his last step before touching the base; generally interference is not called. The runner has to step into fair territory to touch the base that is in fair territory.

 

Runners on 1st and 2nd, 1 out.  Batter hits a fly ball in the infield that the umpire deems catchable.  Ball is not caught and runner at 2nd safely advances to third.
Correct ruling:  Batter is out.  All other runners are safe.  On an infield fly, the runner only has to tag up if the ball is caught. 

Baseball-Rules.com explanation:

Infield Fly Rule
The only difference between an infield fly and any other fly is that the batter is out when it is declared, and the ball does not have to be caught. Because the batter is declared out the runners are no longer forced to run, but they may run if they wish, at the risk of being put out. If the ball is caught they must tag-up before running, the same as on any fly ball. If the ball is not caught they may run without tagging up, the same as on any fly ball.
If the umpires forget to call an obvious infield fly, the rule is assumed to have been called and the batter is still out.
If a fly ball first lands untouched on foul ground before first or third base and bounces untouched into fair territory, it is an infield fly because it is now a fair ball and the batter is out. If the fly ball first lands untouched in fair territory before first or third and bounces untouched into foul territory, it is just a foul ball.
The Infield Fly is a judgment by the umpire that the ball could be caught with ORDINARY EFFORT by a player who was stationed in the infield at the time of the pitch. It is not automatic just because it's a pop-up in the infield.
Rule 2.00 Infield Fly Official Baseball Rules

Situation #4 

The count is 3-2, the ball is pitched and the batter purposely leans into the pitch to allow it to hit him.

Correct ruling:  The ruling depends on whether the umpire detemines that the ball was in the strike zone when the batter was hit.  If it was in the strike zone, the batter is out, because it is strike 3, since the batter made no attempt to avoid the ball.  If it was not in the strike zone the batter is, officially, awarded a walk, not a hit-by-pitch, again because he made no attempt to avoid the ball.

 

Explanation from Major League Baseball Official Rules, Rule 6.08:

The batter becomes a runner and is entitled to first base without liability to be put out (provided he advances to and touches first base) when --
...(b) He is touched by a pitched ball which he is not attempting to hit unless (1) The ball is in the strike zone when it touches the batter, or (2) The batter makes no attempt to avoid being touched by the ball.  If the ball is in the strike zone when it touches the batter, it shall be called a strike, whether or not the batter tries to avoid the ball. If the ball is outside the strike zone when it touches the batter, it shall be called a ball if he makes no attempt to avoid being touched.
APPROVED RULING: When the batter is touched by a pitched ball which does not entitle him to first base, the ball is dead and no runner may advance.