Good questions Sam. I'll try to answer them, but hopefully the always eloquent DcS will provide some additional insight.
1) Yes, dental students go to basketball games, as most everyone in Chapel Hill is a hoops junkie. All students - undergrad, grad, and professional - get tickets in the same way. The week of a ticket distribution, everyone who wants tickets has to pick up a numbered bracelet at the football stadium. That Friday, a number is drawn to determine the starting point in the line. The next morning, everyone with a bracelet has to be in line at the Smith Center by 7:00, and the distribution commences. They give out about 6,000 student tickets. DcS can elaborate further. I'm not sure if there are any distributions left for this season.
2) Grad/Professional students do have a higher percentage of parking passes alloted to them, but you have to live at least 2 miles from campus to be eligible for a pass, and even then you enter a lottery. Parking is very scarce in CH and UNC. My personal theorie is that the socialist tree huggers who run the town are trying to completely eliminate cars, but that's probably a little bit off base. Chapel Hill has a very efficient bus system. My advice: Live close enough to bike/walk, get a place on one of the many buslines, or live a few miles out and hope for a parking pass.
3) There are plenty of places available for rent within an easy walk/bike to the dental school, but you'll soon discover that real estate and rent is way overpriced in CH compared to most "college towns." We live in a 4 bed/2 bath apartment 2/3 of a mile from the school and pay $1625/month. I would suggest looking for a roommate and place now, as many of the undergrads are going to start looking for places soon.
4) Unless you're an introvert, it's kind of nice to have a roommate. They provide a nice outlet after a stressful day at school, and there are plenty of places to study if your apartment is too distracting. I lived with two undergrad friends last year, and now I live with three classmates, and both situations have been fine.
Hope this helps.