We used to always stay on the Ped Mall at the Sheraton, but have moved next door to the Vetro. Cost is the same (outrageous) but the Vetro has larger rooms with better fixtures and a kitchen. Booked ages ago though, sure everything is sold out downtown due to it being Iowa's homecoming.
The times I've parked near the stadium was for tailgating, and we left our car there on a Friday night to save the spot. Not sure if that's still legal, but parking in general isn't too bad, plenty of lawns you can drive onto for $20 or whatever.
I've never had a problem with their fans in Iowa City. We usually go out downtown in and around the Ped Mall since that's where we stay. Fridays used to be an absolute madhouse starting around 4-5 pm with crazy drink specials, but last time it was fairly quiet. All the bars are more or less the same on football weekends and are pretty mixed with students and the old creepers rolling into town.
Best part of all is the street food after the bars close. My favorite is Marco's Grilled Cheese but the whole Ped Mall is littered with carts selling various greasy foods.
Iowa City is usually my favorite Big Ten road trip.