re: Paper Tray - Deisgn Q

Dave has said what I was thinking. Routing a stopped dado that is deep enough to deal with wood movement side to side can work, but as Dave said you are better off having the bottom grain direction run side to side to deal with the movement.

I'd cut the stopped dados about 1/4" deep on the router table with a straight "up cut" bit. The bottoms would get the nose end rounded over with a round over or half round bit. If set correctly you'll get a tight fit at the point you indicate.

Benefit of having the bottom grain running side to side is the exposed edge will be long grain and leave a smooth finish.

The side rails can be glued to the bottom panel with glue applied to the first few inches at the front. This will keep the joint tight. The back rail will hold the rear end tight side to side.
The bottom panel can be allowed to expand/contract by having it slide under the back rail and use a slotted screw hole in the center to secure the bottom to the back rail.
You can get a "cleaner" structure at the back by just cutting a deep dado in the back rail that the bottom panel slides into. The depth should be enough so that the panel will not bottom out when at maximum expansion and not fall out at maximum contraction.
Given the dimensions you'll probably need to make the panels from glued up boards unless you can find some that are wide enough.