Whats the basic puzzle structure for building a cabinet type?
Is it better to have the sides go the full height or fit the sides between the top and bottom piece?
Assuming simple edge joints and no dados or rabbets, then I recommend the sides to fit in between the top and bottom panel. I also recommend the back fit inside the rectangle formed by the the top, bottom and sides. In other words, if the cage will be 48" long and 18" high, then the top and bottom should be 48" long and the sides should be 18" minus 2x the material thickness in height. The back panel should be the same height as the sides and 2x material thickness under 48" in length.
Do note that hardly any 3/4" plywood is actually 3/4".
Now, with that said, I'll also go as far as to suggest it probably does not matter a whole lot. Some frameless kitchen cabinets are designed less sturdy than this and are assembled without glue and hold up heavy stone or concrete counters. Granted they are screwed to wall studs and sometimes to the floor, but nonetheless I think many would be surprised at just how flimsy some kitchen cabinets are.
For another example, I built my bearded dragon cage from scrap and broke all of the rules I listed above and did not even use a back that went all of the way up to the top panel (due to the false ceiling). I had to due to the lights I wanted to fit in the cage so the top and bottom fit between the sides. The dragon is long gone at this point and the cage sits empty in my basement and acts as a storage area for our bottled water. I have had hundreds of pounds of water sitting on the cage for over a year now and the joints are still fine. In fact I can have three cases in the dead center of the cage and the sliding doors still work fine. I try to avoid that but my wife occasionally really piles stuff right in the middle. I'm shocked it's held together.
Then, for sticking it all together, what methods are best to use? Screws (predrilled/countersunk) and glue? I have a biscuit joiner thats been sitting on the shelf unused for over 4 years now. I'd kind of like to break it in.
I also have access to a trim nailer able to shoot 2" trim nails. Should I use some/all/none of these ideas?
Bisquits are nice for alignment and do add some strength to a carcass made with edge joints only.
If you really want to avoid screws on the outside or don't like the look of plugged screw holes then I think you can get away with biscuits and brads. Or you could use a kreg pocket hole jig but that seems to be overkill since you already have a nailer and biscuit joiner.
Definitely use the biscuits and brads for the initial glue up. If you're not comfortable with the end result you can always add screws.
BTW, a lot of these decisions may be determined by what thickness of material you choose. If you went down to 1/2" then I'd strongly recommend rabbets and dados.
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