I'll touch on the big stuff here and you can PM me with any specific questions that you have.
I did this trip with a friend 2 years ago in early October, except we went on through to the South Entrance. I posted a trip report on this a few months ago. Several others here have done portions or all of the trip you plan. It is an awesome trip that time of year. Ten miles a day is a fine pace, it will leave you plenty of time to explore a bit as you travel or once you reach camp. The trail was in great shape, the creek crossings were easy and crossing the Yellowstone and Thorofare were also easy with low water, which is typical that time of year. The only real climb is up Lynx Creek to Two Ocean Plateau.
The Park Service requires that you camp in the campsites you have signed up for and if the rangers catch you in other than the assigned campsite, you have a high chance of being ticketed, depending on the ranger and your excuse (in my experience). Usually that late in the season, there are few enforcement rangers about.
Going solo is fine as long as you know how to travel in grizzly country, and understand back country risk management. My wife bought me a SPOT so I can go solo without her stressing too much. I recommend bringing one or some other similar unit. With no disrespect to Jackson, the bears are entering their hyperphagic stage in the coming weeks as the berries ripen. They are no more or less aggressive during this time, just pay attention if you are traveling adjacent to berry patches and through the whitebark pine community in the high country.
Weather will likely be the greatest determinant of the trip's success. Expect lows around freezing and highs in mid-70s. When I was there 2 years ago, we enjoyed beautiful fall weather most of the time but also had a crazy thunderstorm (at night, of all times) with high winds and hail. It had snowed in the Park a few weeks earlier but most of that had melted off. We also had 1 night of solid rain. Last year I was blown off a trip on Madison Plateau/Pitchstone Plateau in early October due to heavy snow. Be ready for anything.
I'd recommend heading west to the South Entrance. Big Game Ridge and the upper Snake River valley are worth the trip. Hitchhiking in Yellowstone back to your vehicle is usually an easy venture, especially at that time of year. Also, have a backup plan or 2 in case the high country is closed by winter storms. There's a ton of lower elevation treks to take that will provide a great experience also.
Enjoy your trip, it's a great one.