Here is a suggestion for your Yellowstone days. As always, just because you have an itinerary doesn't mean you have to stick to it if something more interesting comes up (mama bear and cubs, wolves, awesome geyser eruption, etc). It's just a guide to head you in a direction from your lodging with an indication of what you might see for the day.
June 12 - Today, I would spend most of the day in Grand Tetons, planning to arrive at Old Faithful late in the afternoon. You do need to make sure you get here before dark. If you have time you can do West Thumb Geyser Basin on your way in. It's a good place to stop and stretch anyway. If you get to Old Faithful before the Visitor Center Closes, take a look around. You will probably be able to see at least one OF eruption this evening. You might be able to catch a Ranger program late this afternoon or this evening. Overnight at OF cabins
June 13 - Make this your Geyser Day. Start at the Visitor Center, perhaps do the Ranger-led walk at 830. Walk around behind OF and down to Morning Glory Pool (vault toilet down here) on the boardwalk and back up toward Old Faithful on the paved walkway. Check at the Visitor Center and see if they have any eruption times for the predictable geysers. After you finish here, hop in the car and head out to the Loop Road going North. Stop at Midway Geyser Basin, drive Firehole Lake Drive, pick up Fountain Paint Pots (the exit from Firehole Lake Dr is right across from the entrance to Fountain paint Pots, but you can't go straight across. Turn left or right, go to the next pull-out and turn around.) and head toward Madison Jct. Just before you reach Madison, turn left in to Firehole Canyon Dr. which heads back south for a couple of miles. Rejoin the Loop road and make your way to whichever place you have decided to stay, either back to OF or on to Canyon. In any event, there are flush toilets at Madison Jct so definitely stop there if you are headed to Canyon. Overnight at OF or Canyon (I would probably go with Canyon)
June 14 - Make this your Canyon Day. Stop at the Visitor Center. (Nice flush toilets, lol) Head south on the Loop Road, doing the North rim viewpoints (if only one, I like Brink of the Upper Falls) and the South rim (Artist Point is the easiest access on both rims.) Return to the Loop Road. Go South through Hayden Valley stopping if you see people out with cameras with long lenses or spotting scopes. Continue South picking up Mud Volcano, then go to Fishing Bridge. Park and walk across Fishing Bridge as this area is rich in wildlife. I’ve been able to spot otters, golden eagle, muskrat, white pelican from here. Head back north and do LeHardy Rapids, then north again thru Hayden and back to Canyon. If you want to do a hike in the area, there are several plus a number of Ranger-led hikes. Overnight at Canyon
June 15 - If you want to do Beartooth, this would be the day for it. Head north from Canyon over Dunraven Pass. At Tower Roosevelt Jct, turn right and go out thru Lamar Valley, then out the NE entrance. Head up over Beartooth and either go to Red Lodge for lunch or make your turn around point at Rock Creek Vista. Retrace your steps. I would probably not stop at Tower Fall or overlooks (unless it is wildlife watching!!) within Yellowstone on your way to Beartooth Highway, but would pick them up on the way back. Overnight at Canyon
June 16 - Today I would head North over Dunraven again, doing Tower Fall area today if you did not get to it yesterday. At Tower Roosevelt Jct, head toward Mammoth. Drive back in to Petrified Tree (sometimes some black bear back in here) and do Blacktail Plateau Dr (this is dirt/gravel, so you want to take in to consideration your rental car contract). Go on to Mammoth. The best bathrooms are the new ones next to the bus parking lot and the gas station. Stop at the Visitor Center. If you are interested do the walking tour of Old Ft Yellowstone, then head up to the Lower Terraces. From here, head South toward Norris. As you climb up out of Mammoth, make sure you do the Upper Terrace Drive on the right. Head to Norris, stopping here if you have time, as well as Artist Paint Pots, again, if you have time. Overnight at West Yellowstone (I agree with Tet that West Yellowstone would be better for your last night in the area rather than back to Jackson)
Have fun and let me know if you have questions!