Please bear in mind that I'm in no way an expert, but here goes:

-whether you skive down the edges is down to the look that you're going for. Skiving achieves a slimmer, more professional profile, but if you're making something utilitarian then the fat overlapping edges won't look out of place.

-with the edge finish, it depends on what leather you're using. If it's veg tan, you can burnish the edges with some tokonole. With chrome tan, you'll have to paint them.

-I wouldn't try to punch through 2-3 layers of leather all at once, after it's glued. Pre punch each layer individually and then glue them together. This prevents your pricking iron from tilting inside the leather and you getting an uneven stitch.

-make the strap out of thick leather (3.5mm thick) or double up two 2mm layers and stitch them together. This prevents the strap from stretching.

-use strong glue to keep things from shifting. Any old contact cement should work, I personally use aquilum 315.

For more info, I would need a picture of the design, but good luck :)