No, there are no lockers at the convention center. You are responsible for your own personal items. For the Arena events, they will be checked at the door of the Arena, which means they'll probably each go into a plastic little bag and you'll have a claim ticket # and will have to wait to get your item back afterwards.
I checked with my D23 Expo rep and he unfortunately did not have a clear answer as to just what will be taken at the Arena door. my unofficial advice is just to wait until you're there and decide what's best for you. you might be able to get away with keeping phones on you in your pocket/purse/fannypack/whatever. If you have no intention of taking photos/video during the presentation, you shouldn't have much to worry about.
as for taking photos without the flash, as far as I know, the only place this applies is in the Treasures of the Archives exhibit (which initially forbid all photography). There are occasions where photos aren't allowed at all, but when they are, the flash should be allowed except for the archive exhibit.
As to how to take decent photos without the flash, it kind of depends on a lot of things. the key response is that there really is no magic way. reason being that in order to take a photo, the camera opens the shutter to let light through and create the exposure which is basically your photo. the primary idea behind a flash is to create a strong enough light source on the subject so that the shutter doesn't have to be open long at all which means you should get a really sharp image. when you don't have enough light, the shutter needs to remain open long to properly capture the image and in that case, it's subject to blur if either the camera or subject moves since it's a composite image of everything the camera sees while the shutter is open.
so now you have a lot of factors to contend with, starting with the room lighting. if a room is lit well enough, you won't need a flash. even in automatic mode, it's possible the camera can register a good photo without the flash. what you want to do is make sure you're holding the camera steady as possible, there are a few techniques that are advised for doing this, but i'm never sure what they are, something about either inhaling before and holding your breath while taking the shot, or exhaling first. I really dunno. but you can simulate different lighting situations at home and practice. plus if you're digital, i advise taking multiple shots every time anyway. you might end up with 3 blurry photos and 1 good one and all it takes it a few extra seconds of your time.
if you have manual control over your camera, you can also adjust the shutter speed and other things yourself and just experiment. in my personal experience, I can probably get decent clear shots as slow as 1/20 of a second at 1600 ISO. the problems in my experience have all been with automatic mode in which the camera over-compensates the time needed to take the photo without a flash.
the ISO is the other primary factor. the lower the number, the higher quality the image, but the more light/time is needed for the photo to be properly registered. it's comparable to film speed on film cameras, so 400 or so is generally ideal. the higher you go, the more grainy your images can get, especially in low light situations. but again, the lower the ISO, the longer your shutter must be. and that all depends on your lighting. you can go outside on a bright day and use ISO 100 and a shutter speed of 1/8000 and get a great photo, but if you tried that indoors, you'd probably get an all black image from not letting enough light in.
if any of that makes any sense. basically it's trial & error and taking as many photos as possible hoping a few will turn out right. that's my most honest advice.
also remember they are selling a limited edition collectible 'catalog' of the items on display at the exhibit. so that's bound to make a great keepsake with photos better than anyone could possibly take on site.