I’ve only been taking trapeze / aerial dance classes since the summer and am nowhere near performance level, but I’ve had a lot of the same questions.
For entering the straddle position, I think it’s a combo of both shoulders and core, especially to get the legs in the right position. I have a weak core relative to my arms and back, and while I can enter hip/knee hangs relatively easily, I can’t get into a straddle yet without a lot of flailing (the straddle is also how I coined my signature move ‘Dead Monkey Falling from Tree’). I think my lack of core strength accounts for this. My teacher said to build these muscles and to get used to the straddle pose, lay down near a poll (like a bed post) that you can grab and lift both legs above your head and into a roll of the back, but by pulling with your abs as much as possible, and then slowly lowering them. (I have no idea if that makes any sense in writing, but you’re essentially doing the straddle motion on the ground with abs engaged.)
For just a pull-over hip hang, it’s almost all shoulders and upper back for me. I asked my teachers about it, and they said just basic ab exercises, push-ups, and pull-ups are the most important, with emphasis on the pull-ups. (She also said yoga and pilates helps). We also do a lot of banana roll sets, and if you have time with the bar, crunches while hanging.
You can buy pull-up bars relatively inexpensively at sport’s stores, so that might help if you have a place to mount one. I’m also looking forward to what others have to say. Good luck with your progress though! I love the aerial arts so much.