East Gate close at 8pm I believe. All other gates open 24 hours.
There is also speed limit (45?) in the park, strictly enforced. We were stopped by ranger once at Lamar Valley for driving 10 miles above limit.
The section from Yellowstone Canyon to Tower is very very steep, narrow and extremely winding.
Road to Mammoth Hot Spring is also winding and steep. But well worth the drive up to North Gate - you see the historical stone gate when Yellowstone first set up. Other roads are fine.
I would suggest you have at least one night stay at Gardiner,MT, just outside North Gate. Work the day into your plan as you would not want to skip Mammoth Hot Spring area anyway. The Hot Spring has dried up but the area is still well worth to see. Most likely you will see herds of Elk at Mammoth Hot Spring town too.
Inside Park, you can find good size hotel room for your family in the range of $200+ in the Yellowstone Lake area. Old Faithful also has a newer hotel that have good size room. No way you would pay the same price outside park versus inside park. Inside park for same accommodation (yes, they do have it, in new construction) would be about 70% higher if not doubled.
Inspiration point at Jenny Lake deserves the hiking if you have time and energy - I believe it is 4+ miles.
Page and Monument Valley area - the roads are fine, you can drive avg 65mph no problem. However, going to Gooseneck you would be driving on some secondary roads, and the spur road leads to Gooseneck is single lane (coming and going), extremely winding but rather short distance (a few miles ?) It is just for getting to Gooseneck.
I still think you should detour to Lee's Ferry - that is the beginning of Colorado River and before the bridge at Glen Canyon was built, it was the only place peolpe cross Colorado River. We were glad we did, though it also meant a very long day that day.
Hwy 89 at high altitude is winding.
Zion and Bryce also have narrow winding roads. Zion is at least a drive thru. Bryce is a pain at the butt backtracking route.
The unforseen would be, road congestion and weather, and sickness. Otherwise, I think you will be OK.