I've been looking to get out to Lyell myself and have been scavenging any information I might get over the last week or so. Here are my best answers for you:
1) The most recent trip report dates back from last week-end (June 16-17) and can be found at http://yosemitenews.info/forum/read.php?3,56832
. As you can see (especially from the last picture), the trail is mostly snow-free, although muddy, until you get near the Lyell-Maclure saddle. There seems to be plenty of snow on the saddle and Lyell Glacier. A summitpost user also posted a zoomed picture a while ago (dated June 2nd): http://www.summitpost.org/phpBB3/north-peak-conness-conditions-t62321.html#p878364
. It is always recommended, although not always necessary, to bring crampons and an ice-axe for Lyell, but looking at the picture and knowing that it is still early season, I would say that they will be very useful for this week-end. They will also offer you a greater number of options once you get there... No word on the snow condition there, but I think they would good assuming you get there early enough.
2) There are no established campgrounds along Lyell Canyon. It is a backpacking destination with dispersed camping. As such, there will be no bear lockers and you will have to carry and store your food inside a bear canister.
3) There is plenty of water all along the trail and many lakes. I don't think that water will be a problem. You get to choose your camp, so pick one with easy walking distance to a water source (remember you have to camp at least 100 feet from water sources).
Did you get wilderness permits for your trip? Will you be going for a 1st come 1st serve permit? My main constraint for climbing Lyell is getting permits for it and I have been pessimistic getting a permit for a week-end. I would love to get some beta on the 1st come 1st serve (week-end) permits...
Hoping this helps...