Headlamp: my current go-to is the Black Diamond Sprinter. I wanted to get a rechargeable one because I told myself I'd never have dead batteries, so I always left with brand new batteries. Buying new ones before every trip when the old ones were probably good seemed wasteful. I used this for my hike up Basin in December. Total time on was probably four or five hours with maybe an hour and a half in the morning and three hours in the evening. I don't know what the temperature actually was, but it was single digits at best and maybe negative single digits at elevation. I never took it off between morning and evening (but turned it off). That was over thirteen hours outside in the cold, and it still worked like a charm when I turned it back on. Recommendations are a funny thing; a lot of gear stuff can be very personal. But I'm happy with my running headlamp when hiking. I also use it running. 40 mph in the open is no joke! I've had wind that bad only once, on Haystack. It felt actively dangerous despite the terrain not being steep or tricky. I really look forward to seeing the other things you do! This all looks fantastic. It's definitely a learning experience, and I bet even the super experienced people learn things each time. Even the same trail is different on different days.
Funny, we have the same headlamp. It's an amazing one, especially for running, but I would prefer a headlamp with greater on time at max lumens than the sprinter (which I thought was four hours but sounds like maybe that isn't the case), and with more of a flood than spot focus. That sounds like a mighty cold hike at Basin. Even on Mount St. Helen's we didn't experience below zero temperatures.
Funny coincidence! That's why gear can be so personal. What I like you don't. I'll take the cold over wind, though! Cold just means bundling up a bit and maybe slowing down to avoid sweating. Wind can feel dangerous.