1- Don't know
2 - 1st or 2nd does not matter, but be sure to do it. This is my favorite entrance to the park; granted I lived in Jackson, so I am partial, but driving through Teton into the park is a nice drive. You could either stay in Teton, or close to it via night or two via Colter Bay, Gros Ventre or others which will allow you to spend some time in Jackson, Maybe take the Tram in Teton Village, and spend a bit of time in Teton. Don't treat this as a day trip while "on your way to Yellowstone", as you can easily spend two days or more here depending on what you want to do. Either way, you will find yourself stopping many times while "driving to Yellowstone". If you have more specific questions about Jackson or Teton, let me know. It is worthy of its own agenda, that is for sure.
3- This is a very tough question to answer as it depends on what you want to do/see and where you will be staying. As someone said, Memorial Day, through Labor Day is the height of the season, but also best chance for good weather and any activity you may want. Mid to late September is not as busy with early October potentially being a good time per the changing colors, animal activity, minimal crowds, etc. (same as mid to late May though snow will still remain in areas and not as much color). Here is a couple links that provide closure info...
https://www.nps.gov/yell/planyourvisit/ocd_locale.htm https://www.nps.gov/yell/planyourvisit/parkroads.htm Also, winter is a time that is beyond words. Very few people, though many areas and roads are closed. You have to hop on a snow machine and do this once in your life. Nowadays a guide is required, but still quite cool, uh cold; especially late Jan/early Feb. May want to get there before or after as it can be -30 for weeks, not days or hours as some might be accustomed to... That's just too cold for any activity, including being on a sled.
4- Black Hills? Only if I have time to Kill. Who knows, maybe Trump will be under construction... Nice place to see once I imagine if you are driving through, but if time is constrained, no chance it is taking up valuable time over N/W Wyoming. Not much in the US can touch it. Again, I am partial, but I have also been over most every inch of it.
5 - Yeah, I really don't like to stay at the entrances to the Park, but if you must, West Yellowstone would be my choice during the slow season anyway. It is the closest to the majority of site, though Red Lodge can be good if you want to hit Lamar and Canyon. I don't know how many nights you plan at each, but West Yellowstone could be the majority as the Upper Loop will take less time unless you are really trying to see some wildlife in Lamar. Either way, I would prefer stay in the park over outside since it wastes much time. If full hookups keep you out of the park other than the parking lot that is Fishing Bridge, West Yellowstone has some pretty good sites that are much closer to your activities in the park