I usually go to YNP a couple of times a year, so I'm very familiar with the areas that you are talking about. You have indeed picked the best route for spotting animals, and you are wise not to travel at night. There are no lights, not even at the intersections. Have you ever seen the front of a car after it hits a buffalo? I have. (It wasn't my car, fortunately!).
Having said that, I have never heard of Fireside Falls. Do you mean the Firehole River? If you would like to swim in warm water straight out of the hot springs, the Boiling River (it's actually a stretch of the Gardner River) is much friendlier, and closer to your route as it is right there in the Mammoth area. It's about a half mile walk on flat ground. The Firehole is a little scary, if you ask me. I woudn't go in there unless I was very comfortable in the water. The Boiling River is an area where you can safely soak out of the current and it's a lot of fun. Also, the cliffs around there are good spots to look for bighorn sheep.
If you don't have a good pair of binoculars, please get them! You will definitely want them for this trip.
I would recommend you visit the Boiling River after you check into the Mammoth Hotel on the 8th. Also, go to the cafeteria and place an order for a box lunch to pick up the next morning ( be sure it's ready early!). You don't want to waste time in a crowded cafeteria, and the picnic areas are beautiful! (Also make reservations for the dining room at Mammoth, Canyon and Lake- now).
Then you can get up early (you want to be on the road no later than 6 am. but earlier is better) and drive down to Roosevelt, which is on your way to Canyon. Turn left (east) and drive out to the Lamar Valley. This is the premier wild-life viewing area in Yellowstone- the wolf watchers will be out in force and usually are very friendly, just ask if you can peek through their spotting scopes. If you see the white Yellowstone Association vans, stop and ask where the wolves are- they always know! When you get to the Pebble Creek campground turn around and head back.
Once you're back at Roosevelt, turn left (south) and the road gradually ascends. This is bear territory- if you see cars pulled over (especially with a ranger pressent) there could be a bear sighting. Don't waste time pulling over to look at elk or bison they're all over the place, you'll see a lot of them in three days! It's not safe to look at them up close except from inside a car, though, so if they're right on the shoulder of the road, that might be worth stopping for.
Continuing on, just before Tower Falls there are pullouts where you can see bighorn sheep across the Grand Canyon of the Yellowstone. Then it's up over Dunraven Pass and Mount Washburn- there might be grizzlies lower down on the mountain and sheep & the rare mountain goat up higher.
Then you can check out all the sights around Canyon in the evening. The next day get up very early again and look for animals in the Hayden Valley. You will arrive at Lake fairly early in the day- there is a nice walk around Pelican Point, if you go do take some bear spray, though, or go with a group. The restaurant is nice, you will need reservations, I'd make them now. I honestly would not bother with a trip out on the Lake. The ride at Jenny Lake in GTNP is better (and it's also the only way to get across it!).
The next day, stop at West Thumb Geyser Basin. This is a very nice basin with many different thermal features. Go early before it gets hot. If you want to see Old Faithful, now is the time to detour over there and take a look. There are neat geyser basins to see, as well. Then double back and continue your trip down to GTNP.
You will have a blast, and you are minimizing your driving. You can't see all of YNP in 4 days, it's best to get as much out of the places you CAN see- and remember, early to bed, early to rise is the way to see animals and avoid crowds.