How are broken bats dealt with in baseball (MLB in particular)

Question

If the batter Bob hits a pitch, and his bat breaks on the hit, how is that dealt with?

In particular, I'm concerned with 4 scenarios:

1. the broken piece of the bat just flies off, apparently affecting nothing
2. the bat breaking ruins what would've probably been a home run, and the batting side ends up with a significantly worse result (e.g. a simple infield hit).
3. the broken piece of the bat flies off and hurts one of the fielders (or pitcher) badly
4. the broken piece of the bat flies off and not only hurts a fielder, but hampers what would've been a relevant play by that fielder. E.g. a fielder Fred could've caught a fly ball, but the broken part of the bat flew into him, hurting him and hampering him from catching the ball.

Lastly, I ask if there are any penalties for constantly having your bat break during games (e.g. purposefully bringing compromised/cracked bats)?

Answer 1

As mentioned, most of this is covered in the MLB rulebook, specifically the comment on 5.09(a)(8):

If a bat breaks and part of it is in fair territory and is hit by a batted ball or part of it hits a runner or fielder, play shall continue and no interference called. If a batted ball hits part of a broken bat in foul territory, it is a foul ball.

1. If the broken bat does not interfere with the play unlike what's commented above, play shall continue.
2. The end result of the play is the end result of the play despite what may have been if the bat didn't break.
3. The comment above covers this scenario.
4. The comment above covers this scenario.

And lastly,

• The penalty for purposefully bringing compromised bats is an out, an ejection, and is subject to further penalties to be determined.

Rule 6.06(d) covers compromised bats. A cracked bat isn't always a compromised bat but a cracked bat can uncover a compromised bat - as with Sammy Sosa in 2003.

6.06 A batter is out for illegal action when—

He uses or attempts to use a bat that, in the umpire’s judgment, has been altered or tampered with in such a way to improve the distance factor or cause an unusual reaction on the baseball. This includes, bats that are filled, flat-surfaced, nailed, hollowed, grooved or covered with a substance such as paraffin, wax, etc.

No advancement on the bases will be allowed and any out or outs made during a play shall stand.

In addition to being called out, the player shall be ejected from the game and may be subject to additional penalties as determined by his League President.