I can speak to a couple of your questions:
1) When training in general, the more specific your training movements can be to the chosen activity, the better. Judging by your descriptions, it sounds like you’re aiming to climb alpine rock (instead of ice/mixed alpine whereby you’re using ice tools). If that’s the case, then I’d say varying the style of pull-ups will offer benefits for you. You can see some pull-up variations in this link:
For the towel pull up, try 1 arm on the pull up bar, 1 arm on the towel for an offset pull up. That’s a fun one 🙂
Also depending on the difficulty level of your rock climbs, you may consider factoring in some hangboarding.
2) Again specificity will be something you’ll want to consider when adding/modifying the workout program. Also, remember that more is not always better. You only have 24 hours and spending that time on more sport specific training will yield you a better “return on investment”
3) Deferred to others.