I'll put the questions to the top for ease:
1) If we start at the museum and go down by the cable car, how do you folks get from the tram station back to the museum? Is there a shuttle? Or just walk?
2) Does anyone know how the conditions at the summit are? Would we need crampons? I'll call the rangers tomorrow, but if anyone has insight, that'd be great. Neither of us have much experience mountaneering or hiking in ice, so if it looks too icey, might not go to the top. Safety first!
Thanks so much to everyone for replies! I lost track of this because for some reason the website stopped notifying me when I got replies, so I assumed nobody was saying anything. Appreciate it!
Well, the cat's out of the bag. My honey and I were discussing the hike last night and as she was googling it, she came across this forum and my thread (good call, Perry). Foolishly I used my real name thinking she'd never come across this! It'd be a funny story to tell the kids. At least now both of us will be in anticipation on the hike

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Florian, I like the suggestion of going to the East side of the hut on the summit, if that's more secluded. If I get inspired earlier on the Skyline trail, I'd do that.
We got a campsite reserved at Tamarack, so no hurry to get back down. So Cornell (thanks, Afaraday) might be convenient if we decide not to do the summit on day 1 and do it in the morning. Miller Peak could be cool too, thanks, Guest.
Halhiker, agree the peak will be awesome and a place for grandkids to come back to

. Though for me, the proposal is something deeply intimate and personal, so I will try to stick to something a bit more private.
CalssVTony - love the wedding pic! I think ours will be at one of the ski resorts in Utah, off-season.
Thanks, all!