From the frying pan into the fire! From straight lines to perfect circles. Circles are tough - at least for me - so be prepared.
You want to use a hardwood (cherry, poplar or something like that) because of the delicacy of some of the cuts. When I was starting out, I found 1/2" to be the easiest. It is thick enough that you can use a #3 blade for the finer cuts but it can handle the 1/2" with no problem. Alternatively, you could stack cut with two pieces of 1/4" wood and make two at once. 1/4" inch would probably look a little better if you want to hang it.
You could use a 1/4" piece of plywood instead of a second piece of hardwood if you wanted. It will probalby chip out in some of the delicate places but it is meant to be a sacrificial piece if you use it (a second piece of hardwood would not be sacrificial unless you wanted it to be). If you use plywood, make it the bottom piece because the bottom piece gets the most fuzzies.
You're right on with your thoughts about the cutting order. The final cut would be the outside of the piece with the teeth. You might want to reinsert and tape down some of the large internal pieces before cutting the outside. It'll give you a place for your fingers to rest on while cutting and reduce the risk of breaking the internal parts while you cut the outside. That is optional, do it if you think it will help.
