First it’s supposed to have a a disguise mechanism, it can change shape and take the form of an ordinary object to blend in with the surroundings. But the mechanism broke, and it’s now permanently stuck in the form of a police box, that’s a a telephone box for policemen used in the early to middle 20th century before they had radios. (The telephone box time machine in the Bill And Ted movies is a direct parody of this)
Second, it’s bigger on the inside. On the outside it’s a small box , but inside it’s got a large control room and a lot of corridors and rooms, even an indoor swimming pool.
Why prepare? The Doctor rests secure in his ability to improvise.
Of course if the Doctor had had a cricket bat about his person, he could have pulled a similar stunt but would have had a larger object to use as reaction mass, which would have accomplished his ends much more easily. In other words, instead of being a bowler or a fielder, the Doctor would have been a batsman, if he was prepared.
Nah, the force would be way out from the CG, so most of the momentum would be converted to rotational momentum, sending the Doctor spinning but with little linear motion.