I answered this in a PM to James, bit I'm also going to post the response here in the thread in case it has value for anyone else in the future.....
Sorry for the delay in responding. Its been a busy couple of days. First off, I'm kinda envious over your trip. I really miss living in Star Valley and Western Wyoming in general. We lived just north of Etna and I dearly miss being able to just walk out the door and vanish for a day or two in less then ten minutes, lol. My wife is still an employee of the hospital in Jackson (works remote in Accounting) and she gets to go back once or twice a year for in-services, but I haven't had the chance since sick parents on both sides forced a move back down here in 2015. ME making the trip at some point is on the bucket list, though if I can ever make my work schedule line up with one of her training trips.
Anyway, as far as the Grey's River loop, just doing the straight through drive from south to north (Smoot to Alpine) generally takes 4-5 hours, but you can take as long as you want to run it. Honestly, you can drive pretty much all dirt from I-80 to Jackson and Pinedale if you wanted to take the time to do it, but in June there could still be areas that are closed or snowed in with down trees depending on how deep the snow pack was this winter. Grey's is generally open around the 1st week in May, so it should be ok by the time you get up there.
There are numerous pull offs where you can set up camp for a day or three if you can find a spot that a local hasn't left their trailer to "claim" the spot and, unfortunately, it is pretty common for that to happen. The FS does enforce the 14 day limit, but it can be spotty if the ONE LE Ranger is busy, because they've got a big area to cover. There are also three "developed" campgrounds along the route, those being Forest Park, Moose Flat and Murphy Creek IIRC. At least one has potable water and one or two charge a nominal per night fee. I don't remember the details right off hand, but if you go onto the Bridger-Teton NF website there should be links to the campgrounds. I used to just boondock in the pullouts if I wanted to spend a couple of days in the woods vs. at the house on my days off. I didn't really have any "special" spots. The whole area is beautiful and you really can't go wrong no matter where you decide to stop.
Once you come out in Alpine, you'll be about 45 minutes from Jackson and another 35-40 from Yellowstone. Its all pavement, so the only hangups would be the amount of traffic headed the same way. There are also some beautiful pull outs along 89 as you go up Snake River Canyon from Alpine towards Hoback Junction, so be sure to stop and take in the sights if time allows.
The main thing is to just be prepared for the unexpected since forest road conditions in the whole are are dependent on how bad the winter has been. I do recall there was a landslide near the Alpine end a year or two ago, but I'm pretty sure they were able to reroute around it and the road was reopened. Regardless, have a blast!