Wood – While building a bed frame, at what point should I start worring about cross beams

The simplest way to do this is with a simple miter joint. Miter all sides of your panels at 45°. You'll need a tablesaw to do this, or a very accurate circular saw with a straight cutting jig (i.e. a track saw). To visualize it, each panel will end up looking like a squashed flat top pyramid if you lay it down on its outer face.

It will be very difficult to get the miter perfect, especially if the boxes you're creating are rather large. Any gaps will be obvious since the layers of ply will be visible underneath the veneer. The corners will also be fairly sharp and fragile since you can't sand or rout them down without exposing the plies underneath. If the biggest dimension of the boxes is 11.25" or smaller, I'd just use a 1x12 instead of plywood.

Then use a square block or beam at each corner on the interior when you're gluing and clamping the corners.

Alternatively you could do this with a router and a router table to create a lock miter joint, and then the blocking inside the box would be unnecessary:

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1. Tools -- as long as you have a circular saw (and a homemade or bought rip guide), you can do this. No tablesaw, no jointer, no planer, unless you really want those. If you absolutely have to buy a new toy, consider a biscuit joiner (Porter Cable 557 is the classic, and readily available used), as that'll make assembly of the cabinet easier and yield a more solid product.

2. Wood -- it sort of depends on the look you want. If you're concerned about the window, I'm assuming children, who will 'distress' a softwood like pine over time. A softwood will be cheaper and easier to work with, though. (Maple is nice, but notoriously difficult to stain well. Birch would be an okay choice. Oak looks like oak, if that's what you'd want.)

3. Considerations -- just make it solid and it'll be fine. Build it a little undersized for the space and use the side/top to fill in the gaps. Design adjustable feet to prevent agony when you discover that your floor isn't level at the front. Google 'french cleat' for an idea of how to lock it into the back wall.

4. Upholstery -- personal preference rules here, but I'd have a separate hunk of plywood with foam and fabric/leather on top. Use a few dowels to index/locate the padding and keep it from squirming around. You could farm that out or diy it... it's not hard, though you have to work a little at getting the fabric stretched nicely. The price will scare you, but I'd go for high density foam for durability.

5. Windows -- I think the extent to which this'll be a problem depends on the exact location of the windows and the inhabitants. If it becomes a real problem, you could screw plexiglass in front for protection.

6. Design -- this is simply a cabinet with different proportions. Start with a separate base made of 2x4s that you can level. (Perfect studs matter here... take the time to pick nice and dry ones.) Then it's just a box (I suggest birch-faced 3/4" plywood; no need to go to town with baltic birch, though) with a base, 2 sides, hopefully a central support or two, a top, and a solid back. (Typical cabinets these days have a flimsy back panel; I'd recommend something more substantial, like 3/4" ply.) The aforementioned french cleat will tie it to the wall. Depending on the visual cues elsewhere in the house, you can leave the front plain or dress it up in various ways. (Google 'applied molding'.)

7. Feasibility -- you can do this. Absolutely. Positively.