Max push-ups, Max plank, Max pull-ups
The no-equipment requirement is forcing the measurements towards strength-endurance tasks instead of true strength. Push-ups require some strength as a prerequisite, and strength definitely affects ones ability to do more push-ups in a limited amount of time, but a maximum number of push-ups is a test of strength, strength-endurance and conditioning, not just strength. The same applies to the plank, and to pull-ups to a lesser degree. As supporting evidence, consider that Dr. Kilgore has muscle-endurance standards, which happen to test the maximum number of push-ups, pull-ups, and sit-ups a person can perform in one set, defined as "not more than approximately 5 seconds pause between reps".
The no-equipment clause is an external constraint that does not assist us in testing pure strength. The only way to get closer to testing strength with bodyweight exercise is to demonstrate gymnastic skills, which of course have elements of balance and learned skill instead of just strength. These strength-dependent skills include the front and back lever, press to handstand, muscle-ups, and others that require rings or other equipment.
The pull-ups could be simply and dramatically improved by making them a one-repetition-maximum weighted pull-up test, out of three attempts. This would require weight plates and a dip/pull-up belt. The other tests do not seem to be closely correlated with strength.
Existing tests of strength
Pure strength is best tested with barbells or machines, as is done in powerlifting competition and scientific studies.
The sport of powerlifting would more accurately be called strengthlifting. It tests strength with one-repetition maximum lifts in the squat, deadlift, and bench press. The person being tested gets three attempts at each. Knowing how much weight a person can pick up off the floor, or push off their chest, or stand up from under, is an excellent survey of their whole-body strength. This takes much longer than fifteen minutes--perhaps an hour or two--and requires access to a barbell, plates, and squat or power rack, as well as someone with basic knowledge of the lifts to verify that each one is performed correctly.
This idea could be extended to any lift. The squat/bench/deadlift is well balanced for overall strength--though it omits a pulling motion--but a one-repetition maximum in many lifts can test strength. I track my maximums in overhead presses, farmer's walks (which involve some conditioning), power cleans (which, being a fast lift, is more a test of power (speed-strength) than pure strength), Romanian deadlifts, push-press, and so on. Strength standards for common lifts can be found at exrx.net, as well as from Lon Kilgore and elsewhere.
Many scientists use one-repetition maximum exertions on leg press or leg extension machines, or test maximum grip strength. In untrained persons, these can be used as rough analogues of overall strength. Sometimes bench press or squats are used, similar to powerlifting.
Feats of Strength
One awesome but subjective way of measuring strength is the completion of diagnostic tasks that are impressive. These are related to strength but often involved a degree of other attributes such as balance. On my list:
- One-leg squat (pistol)
- One-arm push-up
- Handstand push-up
...and many, many more. The test is simply, "can you do it at all?" for any of the items on the list. In his CrossFit gym, Dave Werner uses a similar set of Athletic Skill Level tests (PDF) which includes non-strength attributes.
Starting a strength-measurement regimen
Since you are just starting out with measuring your strength, I would recommend the following:
- Make sure you are able to perform basic bodyweight strength exercises. Being able to do a few dozen push-ups, even more air squats, ten dips, and a handful of pull-ups are fair prerequisites for barbell resistance training. Record the number of sets and repetitions you do, and approximate rest periods.
- Find an appropriate novice strength training program. Starting Strength by Mark Rippetoe and Lon Kilgore is an excellent kicking-off point for barbell training. The wiki provides the bare minimum to understand the program, but the book is a superior choice. It teaches you the fundamental lifts (squat, deadlift, overhead and bench press, power clean) in unparalleled detail, provides a lifting philosophy, and outlines a program. Other resources like StrongLifts are okay as well.
- Track your progression in these basic strength movements over time. In each workout you'll prove your squat and other lifts for a five-repetition-maximum. After several months it may be appropriate to test your 1RM, which you can then return to every year or so.
- Remember that while cardiovascular health is challenged by resistance exercise, metabolic conditioning is a short-lived attribute. It would be quite reasonable to keep running or other conditioning work--sprints, sled drags, hill runs, swimming, rowing--in your program in order to maintain this aspect of your fitness.