Thanks for the question, here are my thoughts:
Warm-up. The burpee part of the warm up is meant to get your aerobic system online and working, this is often indicated by the fact that you start to feel a bit sweaty. 10 is a somewhat arbitrary number, but chosen because in our experience this series puts a person well warmed up state and ready to train.
Core: For the easy exercises, add resistance (weight) until you are limited to the specified number of reps. For side-plank, this is easy, grab a bigger dumbell. For bird-dog use an ankle weight or slip on your mountain boots for weight.
L-sit always feels impossible for about 6-10 weeks and then one day you’ll be able to do them. Keep trying, this exercise is usually impossible more for neurological reasons (your muscles don’t know how to fire that way) than strength reasons.
Pull-ups. I know what you’re talking about, and this can be annoying. I recommend holding your feet in either position (in front is more sport specific) and counting it as extra core training. It won’t detract from the work the arms/forearms are doing.
Calf raises, yes a balancing hand is fine.
It’s best to stick to one good exercise for one training cycle so that you can progress the resistance appropriately. Randomized training is no longer training (thank you cross-fit) because it violates one of the three fundamental concepts which is that training depends on a progressive (slowly increasing) resistance. With strength training we progress the weight. With aerobic training we (primarily) progress the duration, and (secondarily) the intensity. (the other two principles are modulated training load and consistent training stimulus)
Swinging. While we don’t advocate swinging, using momentum, we do outline an Incline (horizontal) pull-up progression on pages 220-221 of Training for the New Alpinism. If you have an e-book version it’s on the page before the start of chapter 8.
Hope that helps.