Am I missing something (or are you leaving out a crucial fact) because to my knowledge there are no weight limits (at least not at the weight you are) to getting into police academy and as for the other stuff, that is not on topic. However I'll answer it: The key to fast Beep Tests/Shuttle Runs is just pushing yourself when you do them and keep on doing them. For cardio don't just run on the treadmill since you aren't looking for a constant run (the way that you mention it), go into the basketball court (or wherever the Beep Test/Shuttle Run is held) and do the actual thing. Now don't stop doing the treadmill, but alternate between the two.
I'm not sure how to help with the 100 m sprints. The only thing I can think of is explosive sets of squats and plyometrics. Finally, the dummy drag, try to simulate something similar to that (even if it isn't that much weight) because it will use more than isolated muscles that are worked on while in the gym so it may not be easy.