Just measured a very old tea-shop style tray, been in use for many years, oak, and the bottom is about 1/4" thick. I don't think it should be thinner.
Simple rubbed glue joint, PVA would be fine.
The main thing about a tray is that the base is the tray, to which a rim is added on top, screwed on from underneath. The base ideally extends beyond the rim (edge rounded off, makes it look thinner, etc) so that the screws aren't too near the edge.
If you make it like a drawer, with a bottom panel in slots, it will be weak. The panel may be strong enough but the weakness is the edge detail. If you do do it that way the base needs to be fielded into the slots, making underneath co-planar with the sides, so that when you put it down the base is contacting the surface. (which is what marcros is also saying above)