OK, here is a suggestion on how to use your time. As always, the disclaimer is that if you see something interesting stop! I am giving suggestions for geology related things that I hope will interest the soil engineer! My pardon if it does not, lol.
Day 1: Arrive Bozeman mid-day. Do a shop for a cooler and supplies and have lunch in Bozeman before you head out. If you need to pick up any guide books for the park, stop at the Yellowstone Association Bookstore which is on the right just before you come to the Roosevelt Arch. (Yellowstone Treasures by Janet Chapple, Geysers of Yellowstone by T. Scott Bryan??) About a mile after you come thru the North gate at Gardiner, keep a look out on your left as the cliff nears the river and just before you come to a small bridge there is a pullout on the left. If traffic is clear, pull in there and scan the hillside for Bighorn sheep. Continue over the bridge, pull in the long paved pullout on your right and take another look at the hillside for sheep. Continue to Mammoth. I would probably skip touring Mammoth for the time being, but if you need a bathroom stop, the best bathrooms are between the gas station and the bus parking lot on the right just before you get to the terraces. Continue south toward Norris. Skip the Upper Terrace Drive for now. As you head uphill, you come to a section of road called Golden Gate. It is cantilevered out from the hillside. Your husband might be interested in stopping at the pull out on the left. You then reach a plateau area. Continue south, turning in to Sheepeater Cliffs. This is a really nice section of columnar basalt here. Keep on toward Norris. I love to stop at Apollinaris Spring (…nps.gov/yell/photosmultimedia/0006a-apollin…) which will be on your left, but the parking area in on the right. No one ever stops here. Continue to Norris, and turn left. If you have time, do the Virginia Cascade drive, then on to Canyon to check in. Overnight at Canyon
Day 2: Today, I would try to make the Storm Point Ranger hike (saw quite a number of wildflowers on this hike in June). In past years it has been at 10AM. The trailhead is 3 to 4 miles east of Fishing Bridge. Fishing Bridge is about 1/2 an hour south of canyon, with no wildlife jams, so I would actually allow an hour. There is no vault toilet at the Storm Point Trailhead, so plan to stop at the Fishing Bridge Visitor Center. The bathrooms are not in the actual VC, but in the woods on the east side of the parking lot. After the hike you might want to come back to the Fishing Bridge General Store grill to eat lunch or you might want to picnic. You can take this time to drive out the East entrance road to see the hoodoos. Also drive up Lake Butte overlook (there is a sign) for the view. Return to the Fishing Bridge junction and head north stopping at LeHardy Rapids and Mud Volcano. (Depending on your time, you might be able to work in Natural Bridge hike which is just south of Bridge Bay and Lake areas.) Head North thru Hayden Valley and stop if you see people with spotting scopes or long camera lenses. Do the overlooks at the South Rim (At least the Upper Falls overlook and Artist Point which is at the end of the drive) and then pick up the ones on the North rim if you have the time and energy. Overnight at Canyon
Day 3: Finish up any Canyon overlooks you missed. Head over toward Norris doing the Back Basin and Porcelain Basin, then around toward Madison. Stop at Artist Paint Pots and Gibbon Falls if you like. (BTW, Gibbon Falls is one of the places where you can see the edge of the Caldera, the collapsed volcano upon which most of Yellowstone lies.) At Madison you might want to head out the west entrance road a ways. I would go about 8 miles which might give you a chance to see bald eagle and/or trumpeter swan if you haven't already seen them. Continue south from Madison crossing the Gibbon River Bridge. Turn right in to Firehole Canyon Dr. Rejoin the Loop road and head south. Depending on what time it is, I might head straight for Old Faithful at this point. Once you get to the OF area, go to the Visitor Center. Write down the eruption times for the predicted geysers on the brochure available at the desk for 50cents. You could also head to the Old Faithful Inn and have an adult beverage on the 2nd floor deck overlooking the geyser basin...or ice cream...or a hot drink if it's turned chilly! Overnight at Old Faithful
Day 4: Start with the Ranger led hike this morning at 830 or so around the Upper Geyser Basin. Continue thru the basin after the Ranger is through. Go at least to Morning Glory Pool. (Vault toilet down here) Work your way back up to the main area. If Riverside or Grand geysers are near to erupting, I encourage you to wait. They are both spectacular! I also like to eat lunch at the Lower General Store which is the first one you come to when you are on your way back from Morning Glory. In the afternoon, get back in the car and head back toward Madison picking up Black Sand Basin, Biscuit Basin, Midway, Firehole Lake Dr and Fountain Paint Pots. The Firehole Lake Dr ends right across from Fountain Paint Pots but you can't go straight across that intersection so you have to go left or right and turn around in the next lay-by or you can turn right and go to Fountain Flat Dr do it and come back to the loop road. It will be clearer when you get there! Plus you will have the map you will get at the gate when you arrive. Overnight Old Faithful
Day 5: From Old Faithful head to Norris. Go back North to Mammoth. As you come down in to Mammoth, stop at the Upper Terrace Dr. Continue down in to Mammoth. Do the Lower Terraces and the Visitor Center. There is a self guiding tour of the Old Fort Yellowstone buildings if you have an interest. They are the old buildings the Visitor Center is in and the stone buildings along that drive. From Mammoth, head east. There are several places to stop, Undine Falls, Northern Range nature hiking trail, Wraith Falls and Blacktail Plateau Dr. I think I had mentioned this for wildflowers. It is gravel and dirt so your rental contract may not allow you to go off road. Use your judgement on this. Finally yoyu will come to petrified Tree. Take a drive back in here. The tree isnt much in my mind, but you can often see black bear back in here. Is this the night you have reservations for the cookout? Overnight Roosevelt
Day 6: Today, I would head east from Roosevelt into Lamar Valley. You have several options today. You can hang in Lamar (go at least to Pebble Creek campground.) Be sure to stop at Soda Butte and walk around behind it to look at the swallows nests all over the back of the butte. If you want you can drive out the NE entrance and drive up over Beartooth Pass. If you do that go at least to Rock Creek. I would turn around there if you have cookout reservations tonight. If you don't want to drive Beartooth, you could do a hike. You mentioned Lamar River trail, also Yellowstone Picnic area trail or Trout Creek trail among several in the area. Overnight Roosevelt
Day 7: Head south from Roosevelt toward Canyon. If they are open next year, you might want to stop at some of the overlooks on the left. There is a wonderful line of columnar basalt across the canyon. Continue south past Canyon and the Lake area. If you want you could do Natural Bridge today if you haven't worked it in another day. Continue south to West Thumb geyser basin, then out the south entrance road to the South Gate. To Grand Tetons
Birds will be everywhere, but particularly along the rivers. Fishing Bridge is often a good spot for watching. Possibilities are Golden eagles, bald eagles and trumpeter swans. LeHardy Rapids is a good place to see water birds and ducks. Nez Perce Picnic area between LeHardy Rapids and Mud Volcano is also a good place to sit and watch for birds.
Let me know if you have questions!