1) If you don't mind mountain passes, go across the Big Horns on Route 16. If you do mind the mountains, go through Casper, WY.
2) Tetons first or second would depend on if you are going before tourist season or after. Weather in Yellowstone will get warmer last and colder first.
3) Go before the weekend prior to July 4th or after Sept 1. A week farther from those dates would mean even less crowds.
4) Most definitely see the Black Hills. Figure at least four days in the area. Also don't forget about Badlands NP and Devils Tower.
5) The only place you mentioned worth seeing Yellowstone from is West Yellowstone. The other places are too far to make seeing Yellowstone practical. You want to be there early in the morning or evening to see the wildlife. Unless you can't stay in the park for medical reasons, I strongly recommend it. I hate Fishing Bridge, but if you need power, than it's a better choice than outside the park. Stopping in Cody for a night or two or three makes sense as there are some neat attractions.
Remember, during the times I suggested, you will not need air conditioning but will need some heat.
Enjoy, I've been twice, last time 12 years ago which is really too long ago. Hopefully this year will be the ticket for me.