I camped out overnight for Dear Evan Hansen, and honestly, the rush line experience was almost as touching to me as the performance. Now, in response to your questions, I'll tell you what I did, and I'll tell you what most normal people do (I'm weird. Hahaha).
1. I walked to the theatre to get in line immediately after leaving a conference. I honestly didn't have anything to sleep on, but I did have an umbrella in case of rain. Luckily, it didn't rain! Most people bring folding chairs/lawn chairs, blankets, sleeping bags, sleeping pads, things like that. You're going to be on concrete, so bring whatever you think you're going to need to be comfortable on concrete.
2. I can't think of anything to not bring, but if I do, I'll get back to you!
3. I personally actually didn't sleep at all during the night, just because I had work and such I needed to get done. Sometime around 4:00 AM, I went to the McDonald's on 8th Ave. to get some hot tea because I started getting cold (because I didn't have a blanket. Hahaha). But no, I was not sleepy at all during the next day. However, I can't speak for others because I'm very used to all-nighters and not sleeping much in general.
4. I actually did see two shows that next day! I was fortunate to have been first in line for the Dear Evan Hansen rush (SRO), so as soon as I got my ticket, I ran across the street to get in line to get rush tickets for Bandstand. I managed to get secondary rush tickets, and I loved my seat location! I saw Bandstand for the matinee show and Dear Evan Hansen for the evening show, and I felt completely fine through both. Again, this might not be the case for other people, but I felt totally fine after the all-nighter. Even without a nap during the next day (I had to run to my conference for a few hours in between), I was perfectly alert and present for all of Evan Hansen.
The next day (Sunday), I got rush tickets for Sunset Blvd., and although I only got three hours of sleep on Saturday night, I still felt fine. I got like seven hours of sleep on Sunday night, though!
I recommend that, if you're concerned about getting sleepy or whatnot, maybe you can find a buddy to do the rush line with you so you can take turns going to your hotel to sleep for a few hours. A lot of people do half a night and have someone else take their place for the second half/morning. And depending on how close your lodgings are to the theatre district, maybe you can rush back and take a quick nap between the matinee and evening shows. Or, if you want to try sleeping while in line, and if you're afraid of the sounds and lights disturbing you, maybe you can bring an eye mask and ear plugs or ear muffs. The people in the line are awesome, and sometimes policemen pass by to make sure everything is alright, so you'll be safe!
5. I absolutely have no regrets about any part of the experience! It was my first time, so I didn't know exactly what to expect, but wow, I'm still taken aback when I recall my experience from that night. I met such amazing, kind people, and I'll never forget how I felt meeting and talking with them. I'm in a field of work that has a lot of really bad, rough, nasty people, so I was almost in tears seeing and meeting so many nice, trustworthy, passionate, non-judgmental people all at once. Anyway, I promise you won't regret your experience! I hope the weather will be kind to you (that's the only thing that's unsure, hahaha)! Best of luck, and have fun!
(P.S. If you're reading this and you remember a girl passing out pumpkin bread and banana bread, I just want to thank you so much for that powerful, healing, rejuvenating experience.)