1. IMO THere's no need to divide up bent arm and straight arm work if you are already doing a push/pull system.
Typically, I like full body but if that doesn't work then either push/pull, BA/SA, or upper lower can work. Generally, you don't want to divide into both push/pull and upper/lower because of the decrease in frequency of a lot of the skills you will be doing won't allow good progress.
2a. If your handstand statics are already good then start working them on rings.
2b. SA press should be included with pressing work, after your handstand practice.
3. Depends what your level of ability is... if you're seeing weekly progress I would tend to manipulating your days towards volume vs intensity (or light/heavy or whatever you want to call it). Some variant of DUP is what I feel is the easiest to implement for most people.
4. Prehab elements for all intents and purposes should be done when you feel like you need it. That would generally mean everyday or at least 3-4x a week. Which typically means every training session or if you like doing them on off days instead of workout days thats fine too.