The batter's box provision seems particularly weird.
I understand the desire to speed up the game a little bit, but I really don't see hitters stepping out of the batter's box as a thing that has consistently slowed down games.
The most significant impacts on game speed seem to be a) pitchers taking a long time in between pitches, which really adds up over an entire game, and delays in and around pitching changes and commercial breaks.
There is also c) delays caused by manager arguments and/or replay periods. But, c) doesn't really amount to significant delays IMO and it also seems pretty easy to address in the context of replays, by making managers stay in the dugout (which they've done) and the general speeding up of the replay process which should occur as the league office becomes more practiced at it.
I don't think they accomplish much with the batter's box rule and it actually may slow down the game. It could lead to arguments when it actually gets called, and players who are uncomfortable with it (veterans like Ortiz) might end up asking for time a lot more often before a pitch is delivered.
I don't like the idea of making relief pitchers warm up less on the mound because that might lead to more injuries, as guys don't get the requisite number of warm up pitches. Sure, they can warm up 100% in the bullpen in theory, but in practice managers will call them in as soon as they can and pitchers won't be 100% truthful about their warmth.
The only true and good solution to me is a pitch clock. Hopefully it works out in minor league trials this year. I don't like the big glaring number and the formal clock idea... I would just support more judicial enforcement of the rule much as it currently is. Let it operate like the 5 second rule in basketball inbound plays - umpire discretion. I don't like the idea of formally penalizing a pitcher who by chance starts his motion at 21 seconds after receiving the ball, provided that pitcher is under 20 most of the time.