Variation is good. Just make sure you are in fact hitting the same muscle. For example, hammer curls are not going to hit the bicep the same as ez bar curls; reverse grip tricep push down is going...
Nutrition is definitely #1, but cardio does help reduce fat stores. I'd suggest doing it away from your gym time if possible (i.e. morning or rest days).
Bent over rows, lateral delt raises, anterior delt raises, skull crushers, weighted (dumbells or barbell on shoulders) lunges, shrugs, single arm alternating dumbell curls, pull ups.
Use DBs for incline and flat bench. Should be easier on your rotator cuff. I agree with only needing 1x fly movement. Pick one, rotate them. Dips would be good, too.
Functional style training may be what you are looking for. But the number one thing you need to do to "hang from ropes all day" is to lower your body fat %. Use a calorie counter like MyFitnessPal...
Should be okay. Could also do complementary muscles together then another group in the evening, allowing you to hit the same muscles more frequently but allowing for more rest/off days. Back/Bi...
