I'm going to answer some of your points to the best of my ability. Don't take them as fact, just information and points of view I have gathered over a period of time Do Push Ups, Sit Ups, Pull Ups and Running
If those are the four things that you are going to be marked on, then those need to be the primary focus of your training. It won't matter if you can squat your bodyweight with weights (but weights WILL help, more on that in a moment) if you can only do 10 push ups. Focus on those four elements and train specifically for them -- strength endurance, endurance and cardiovascular fitness.Avoid Training to Failure
For a real life athlete -- such as the military, police, fire service -- training to failure is both unnecessary and counter-productive. What's the use of a soldier if he can't chase down the enemy because his muscles are fried from training the day before? In the military, you need to able to perform day in and day out. Your training should focus on sets and reps that challenge your strength-endurance but do NOT wear your body out to the point of failure.Strength Training to Improve Running and Strength-Endurance
I highly recommend strength training 2 times a week if you can. The focus, as you'll be doing a lot of work with push/pull/sit ups, should be on the two biggest lifts: deadlift and squat. Both of these exercises require a muscle contraction across the whole of your body and will train the legs and lower back effectively - two areas that will likely be neglected from focusing solely on push/pull/sit ups. I can also talk from personal experience that squats WILL improve your running phenomenally! Strength is great in that it carries over well into other areas of fitness -- you will notice the difference in your running once you start squatting. Regarding sets and reps, go for 3x5 and, as said earlier, avoid training to failure.Training Push Ups, Sit Ups and Pull Ups
This isn't something you necessarily have to do but I believe is a good training protocol for strength endurance.
Set yourself a number of reps you want to do for an exercise. For example, "I want to be do 500 reps of push ups every week, Mon-Fri."
500 reps over 5 days = 100 reps a day.
Workout the maximum reps you can do in a set before hitting failure; e.g. 20 reps per set.
Thus, you perform 5 sets of 20 reps each day, Mon-Fri, in order to complete 500 reps that week.
Do the same thing for each exercise, taking into account the level of strength-endurance you have for each (pull ups will likely be less reps; sit ups possibly more reps than push ups).
Every week, assess how you're doing. Are you capable of doing more reps in each set without going to failure? Can you increase the total reps you do every week for each exercise? Are you pushing yourself too hard with an exercise? Take into account these things, assign new numbers for the week, and repeat the cycle.
Oh, and historically, Spartans trained daily Finally: Make Haste, Slowly
A lot of people are always in a desperate sprint to the finish line, whether it's getting bigger, stronger, more endurant, etc. Fitness is a marathon, not a sprint. The human body has a set recovery schedule that does not alter. When you push these boundaries too hard and too often, that's when you plateau or, worse still, run into injuries - short term and long term. Listen to your body. If you feel fried every week, reduce your weekly load of training. A week off from training can boost recovery and give you a hard-earned break. Perhaps your body takes longer to adapt to training - stick to the reps/sets/weight you are doing for another week or two and adapt from there. Your body is the best gauge you have.
Hopefully a few of these things will help you! Don't take them as solid golden rules, use your own judgement and other people's advice to make up your mind on a routine that suits you and your lifestyle
As an idea of a weekly schedule you might want to try, taking into account the above, here's an example:
(Assuming a weekly target of 500 push ups, 500 sit ups and 100 pull ups...)Monday:
Across the day: 100 push ups, 100 sit ups, 20 pull ups. Squat and Deadlifts - 3x5Tuesday:
Across the day: 100 push ups, 100 sit ups, 20 pull ups. Running/Sprint TrainingWednesday:
Across the day: 100 push ups, 100 sit ups, 20 pull ups. Squats and Deadlifts - 3x5Friday:
Across the day: 100 push ups, 100 sit ups, 20 pull ups. Running/Sprint TrainingSaturday:
Across the day: 100 push ups, 100 sit ups, 20 pull ups. Running/Sprint TrainingSunday:
You may prefer to train Mon-Fri and have the whole weekend to rest. The above has a rest day on Wednesday; a mid-week day of rest is often useful