Pushing strength, yes they do relate to each other because all of them are mostly triceps strength. The main difference between various exercises is what other stabilizing muscles are being worked, and how intense or difficult it is.
Ideally you choose an exercise that relates the most to the skill you are trying to develop strength for, such as using pushups with feet elevated to help with a boy's backuprise on pbars.
As for working all 3, depends how intense you need the workout to be. Variety is good, though personally I wouldn't give them 2 exercises with different intensities in the same conditioning session.
Lastly, exercises for the muscles in the back are a bit more tricky to target as the range of exercises is a lot smaller.
My guys use press to handstands, reverse leg lifts, heel drives, heel drives with flutter kicks, as well as arch rocks and shoulder bridges. Keep in mind this is very generalized combining both trunk extension exercises as well as hip extension.
Also keep in mind that different intensities, reps, and sets also target different types of strength. Muscular endurance, hypertrophy, strength, and power. So make sure you giving the right conditioning for what you tying to develop. That is a common mistake I see a lot of coaches make.
Hope this helps?