If you have a chronic shoulder injury then a high volume of pull ups may not be wise. Without know a lot more about that shoulder injury or being a PT I am hesitant to recommend placing too much load or too much volume on it. If your goals are primarily mountaineering then upper body general strength exercises like the pull/push ups are of secondary importance. Taking a wild guess here I’d say that if you have good (for pull/push up) shoulder range of motion, pain free you’d do best with a low volume max strength protocol to avoid overuse injury for high reps. But this is a guess.
As for lower body: Yes, with your PL background you probably have more than adequate general lower body strength. Transferring that high level of general bipedal squat/DL strength to more specific single leg strength will be the most helpful for more athletic goals. Since we generally locomote form one foot to the next in sports as opposed to have both feet planted solidly on the ground during knee and hip flex/extension, hip stabilization is very important. So use the box step however I do recommend you use a bar bell on your shoulders for the box step ups/downs. See this https://uphillathlete.com/strength-demo-box-step-up-and-heel-touch/
I hope this helps.