There are two ways to build the nest boxes, with and without dados. The dadoed joints are stronger and more weather tight but are more difficult to make. In woodworking terms there are three names for a wide cut in a board that can accept another board. A groove is cut with the grain, a rabbit is cut against the grain, and a dado is either that is open to one side. Most people just call all a dado. The two most common methods for cutting a dado are with a router or a dado blade on a table saw. A third method not discussed here is using a dado hand plane. A joint made with the wood just butted together is called a butt joint.
First a brief comment about plywood. Plywood is commonly referred to as 3/4, 1/2, 1/4, etc but the exact dimensions are 1/32" less. Plywood is graded by the finish of the exterior plys and the glue. For the chicken coop an exterior glue is recommended. Plywood labeled ACX has one sided sanded and knots plugged (A), the other side left raw and no plugs (C), and exterior glue (X).
To make a tight joint, the dado must be cut as close to the plywood thickness dimension as possible. A snug fit with a little water proof glue such as Tight Bond II and some brads to hold things in place will provide a weather tight and long lasting joint. A loose fitting dado loses some of the mechanical advantages of a tight fit. A butt joint does not have the strength because the sides of the snug dado provide mechanical locking and support. A butt joint can be as weather tight if made very accurately.
Cutting dados with a router is much easier than cutting on a table saw but it requires another tool and tool bots. For some such as myself an excuse to buy another tool is sufficient reason in itself. there are router plywood bits available that cut a dado the exact width of the plywood. Rather than a 3/4" bit, you will make a much better joint with a 23/32" router bit. One of the easiest ways to cut a dado with a router is with a jig. To build the jig take a piece of thin plywood longer than the anticipated dado and wider than the distance from the center of the router to the edge plus 3/4". Glue a 3/4" by 3/4" cleat along one long edge of the plywood. Clamp the jig down, install the router bit you intend to use for the dado, then holding the router base against the cleat cut off the other edge of the plywood. You now have a jig, just align the cut edge where you want the dado, securely clamp the jig to the work piece, set the depth of the router bit to the dado plus the jig thickness, then cut the dado. I have several of these, all marked with the size of the router bit and the router used. On some I put the cleat in the center of a wider piece of plywood and used a different bit on the other side. A perfect use for all those scrap pieces of 1/4 plywood.
The other common method for cutting dados is using a table saw with a dado blade. There are two common types of dado blades, a stacked dado and a wobbly blade. In both cases the dado blade is smaller that the regular table saw blade, typically for a 10 in table saw the dado blade will be around 8 inches. The dado blade requires more power for the much larger cut, the smaller diameter blade reduces the power requirement somewhat. For Safety when using a dado blade you need to replace the table saw insert with a zero clearance insert. These are available as blanks or you can make your own. Insert the blank in place of the regular insert, clamp it into position, then raise the blade to cut a slot that will have zero clearance and make a much cleaner cut.
The stacked dado blade is just that, a stack of blades. The stacked dado set will have two outer blades and several inner blades of varying thickness. On some sets the inner blades have just two cutting blades while other sets have inner blades with four. THe more cutting surfaces, the finer the cut. For this application, the two cutters will do just fine. You stack up the inner blades until you get the thickness wanted with the two outer blades in place. Shim sets are available to adjust the thickness more accurately. The bottom of the dado is flat with a stacked dado blade.
The wobbly dado blade uses a single blade with an adjustable center. As the center is adjusted the blade takes on more of an angle. In operation the angled blade makes a wide cut. The width of the cut is dependent on the adjustment which is usually marked on the center. The marks are not very accurate even on the more expensive blades so some experimenting and adjustments are needed to get the snug fit wanted. On a wobbly dado the bottom can take on a slight curve, depending on the width set. Most are sharpened to give a flat bottom at 3/4" with an increasingly concave bottom as the cut narrows. In most cases the concave bottom is of little worry but it is something worth checking.
To cut dados (or more accurately rabbits) on the edge of a piece start with a sacrificial fence screwed to the table saw fence. Any straight piece of wood will do if it is about as wide as the fence is tall and about as long as the fence. Most fences have holes for this purpose, if not you can clamp the fence. Set the edge of the fence to expose just the thickness of blade needed for the dado. This is why you use a sacrificial fence as the blade will be running against the fence. Set the depth by raising the blade and cut dados. For dados in the center, you move the fence to make the cut where wanted. The sacrificial fence is not needed for this operation.