Have you looked at Roark ? (the answer to some of your questions should be apparent).
However if you do you'll realise that you can't easily get to where you want to (Roark is giving the maximum for each loading, and these are not (or probably not) co-incident. Have you tried my search term, which should get you to the reference Roark uses ? This should give you a more general description of the internal stresses. Timoshenko (Plates and Shells) is another reference, but a much harder read.
Why fixed edges ? ... because the edges of the panels are supported by stiffeners. But assuming simply supported edges will probably give a higher (more conservative) result.
If I was doing this (and didn't want to use FE on it), I'd analyze a stiffener (reacting 1/2 a bay of panel on both sides) since the stiffener is the key structural element (reacting the bending from the pressure).
If I had to analyze the panels, I might analyze a strip (1" wide, horizontal and vertical) to see how the panel reacts as a plate in bending. Normally this shows that the deflections are large (compared to the thickness) and so the panel reacts pressure as a membrane (with in-plane forces).
"Hoffen wir mal, dass alles gut geht !"
General Paulus, Nov 1942, outside Stalingrad after the launch of Operation Uranus.