1) Expect most of the gully and the chute to be covered by snow. The snow conditions will depend on the weather and time of the day. Definitely bring crampons and ice axe.
2) The weather conditions are hard to predict - I would definitely start monitoring the forecast several days before the trip. Yet, it can still change fast when you are up high. There might be snow-free spots at Iceberg lake. As far as camping gear, it depends on how much weight you are willing to carry and how comfortable you want to be.
3) I personally think the biggest problem with ladges in late may will be route finding, especially on the way back, especially if you pass them in the dark. Don't sprint through this section on the way up and notice where exactly the downclimb is.
4) I am a big fan of snowshoes and I would consider bringing them is there is still significant snow coverage between UBSL and Iceberg lake. I am sure a lot of people will disagree but, personally, I would prefer carrying them up to avoid a mile-plus of postholing. Again, only if snow coverage is still siignificant which may not be the case and if you plan on hiking through this section in the afternoon when the snow is slushy ....
Cheers