what's the furthest distance we should be prepared for?
3-4 miles in beast. You will break up into ability groups based on your initial APFT time upon entry. Fast people will run faster workouts. Slower people will be in a slower group. However, be prepared to run a mile or so in formation to PT locations. Do not be the one to fallout on a run after PT has been completed. Beast is straight forward, showing up in good running shape is a solid way to make your life easier.
Mix up your runs. If you have are trying to improve then try the following.
Do 3-5 mile long distance runs about once a week.
800 m (half-mile) fast paced runs (3-6 sets) 1-2 times a week.
Throw in a 2-3 mile run with other workouts as a warm up or cool down.
100-200 m hill sprints are also good.
You don't need to be the fastest, but being able to stay in formation while moving up a hill is a must.
As has been stated by others, don't just train running. Being well rounded with weight lifting etc. can help improve your running form.
Last thing... go ahead and push yourself on runs. You can sustain a hard pace for longer than you think. Get yourself in a grind mentality.