1. What is the best route to take to Cody Wy?
If you take the 'most direct' route, you will deal with a lot of tolls in Oklahoma. You will also cross a lot of mostly unpopulated country in southern Wyoming. We did not like portions of that route at all.
The route we liked best and also the easiest and most scenic was from Buffalo (I-90) over the BigHorn mountains on US 16 to Worland. There is a shorter route but that involves US 14 and a lot of switchbacks and really tough grades.
From Worland to Greybull to Cody.
If you are going to see the Black Hills, it might be simpler to visit them first and then head west to Buffalo on I-90.
Red Lodge is a LONG drive from Cody, but a beautiful ride on the Chief Joseph Hiway and the Beartooth hiway (US212) Make the drive but it might not be a good idea to try to stay in Red Lodge as a base and certainly not a good idea to drive over the Beartooh towing or even in a large MH.
Again, Cody is a great little town but basing there means you will have a daily drive of a couple of hours to the Park and then that much back.
Suggest you look at Gardiner on the north side of the park or West Yellowstone in the west side. More facilities in W Yellowstone and more to do.
Just my opinions on a couple of things. WHere ever you stay, you will love the country around there.