TL;DR (too long; didn't read) but here goes!
I do what I call "competition binding", that is bias cut, double folded French binding made in a continuous loop method. I sew onto the top of the quilt, miter the corners, do the continuous angle for the joining, and then hand sew with an invisible stitch on the back. When I sew the binding on the top, I start in the middle of a long side, leaving a nice tail to meld together with the other end. Some people can get by with very little, but I need probably 6-10 inches of fabric on each side. I will typically pin to the corner, then fold the miter and pin the next side to the next corner.
I wish I was better at machine sewing the binding and every now and then I'll give myself like a place-mat project for practice but I still prefer look of the hand sewing. I am terribly slow at the hand part and generally get cranky during the process, but instead of swearing I keep repeating "a kiss with every stitch". It works (sort of) for me.
I do generally cut/make my binding a bit wider than many prefer. For me it is the last chance to add fabric to the top. For the "competition" part, just make sure it is nice and full. That is don't trim your top/batting/back to a 1/4" seam and then use a 1/2" binding, trim it to fit your binding.
I used to iron it in half but something I picked up on maybe 5 years ago on this board was not pressing it, leaving it flat and simply folding it as I put it on. It allows for just that slight thread or two movement for the fold and I find I get better results.
I wind it around nice large stiff pieces of cardboard that still fits in my totes, some of it depends on what spare costco boxes I have around, but I like about an 18" piece. You can even count the rounds to have an idea of your yardage. There are charts and such but since I mostly make queen sized projects, I start with a yard and a half of fabric, and make that into a square of whatever size the fabric is (so typically somewhere between 40-44"). Even with a 3-3.5 inch binding, that is more than enough for a typical queen sized project.
For the cylinder idea, I used oatmeal containers many years ago -- but they were a bit sturdier then.