Firstly the big issue for most woodworkers, the question of screwing into the edges of plywood.
You'll see plenty of advice not to do this if you look around, but you'll also see admissions from lots of folks who do it all the same and most will tell you that it works fine if done properly. There are various tips about how best to accomplish this without issues, here's a summary from various sources:
- Thicker plywood is a better choice when attempting to screw into edges. 18mm or 3/4" is often the minimum thickness recommended so you're good here.
- Always centre the screws on the thickness of the plywood.
- Drill suitable pilot holes (note: clearance holes and counterbores are needed in the piece the screw goes through).
- Squeeze the faces of the plywood with a clamp when driving the screw in to physically prevent it from splitting or bowing. This can be worth doing even if using self-tapping screws.
- Don't screw near a corner (even if using the clamp trick). Place screws at least 5-6cm from a corner.
- Pick your screws with care.
- Glue the plywood butt joint, that way the screws aren't being relied upon for strength but are mostly there to initially hold the case together while the glue dries.
So as per the above you should ideally glue the edges of your ply to the case sides but you may want to avoid that for a few reasons.
- the surface plies of modern plywood are often too weak (too thin, and not bonded well enough) to reliably glue to.
- it makes assembly much more difficult and frustrating if you've never done it before.
- much less glue squeeze-out to clean up when you're done (this is surprisingly difficult to do properly, and even harder on plywood surfaces than on solid wood because of the way the veneer is peeled from the log). However, you can protect adjacent surfaces with tape to prevent the problem in the first place, similar to how it's used here to protect joints from finish.
I do not know what length screws to use and the distance between them and dowels.
Longer is better for strength, but increases the potential for the screw to split or bulge the plywood at the edges. But screw type is arguably more important than length. Basic wood screws aren't ideal for plywood, although they can work OK. Confirmat screws are often recommended for use on plywood due to their superior holding power but common chipboard screws should work perfectly well and are far easier to find, these screws are made to hold in material much less dense than you'll find in the edges of plywood so should hold extremely well in anything better.
Once you've selected your screw it would be advisable to do a few tests screwing into some offcuts of your plywood, because the quality of ply varies so much what one person has had success with does not guarantee success for you, using a different make or type. Test to find out what size of pilot hole works best and whether or not you need to clamp the plywood to help prevent it bowing or splitting — the larger the diameter of pilot hole the less the chance of these problems, but at the cost of lower holding power.
Truth be told the dowels you're planning on using as well will add a lot of strength here, much more than they often do in a joint (in solid wood). If you're buying lengths of dowel and sawing it down yourself you should groove the dowel pegs to aid them being driven home, see previous Question, How can I flute my own dowel or create dowel with similar properties. Also chamfer the leading edges, just as is done on commercial dowels, which can be done with a sharp chisel, in a pencil sharpener (yes, really) or simply by rotating the dowels against abrasive paper.