Question 1: I just got into Brazil last week and not only was I asked for insurance by both the gate agents and immigration they both gave it a hard look. My understanding is as of Oct 2nd its no longer needed but I would proceed as if it is. Just a document stating that you're covered should do the trick.
Question 2:
Unlike Mexico where the countryside is very dangeorus, driving through Brazil's countryside is not particularly dangerous, the crime is mainly in the cities, but its just not practical. Parking is a PITA, and ppl drive very crazy. That being side the highway between Sao Paolo and Rio is the best in the country, safe and easy to do, but like @
IstKong said why bother when you can take a $30 flight. SDU is so close to zona sul you can bike there in 25 minutes, maybe 10 minute drive. Its also very easy to catch a bus and you can try to get the executivo seats that recline 150 degrees very comfy, but again flying is cheaper then dirt for that route even last minute. Regarding the open jaw or roundtrip option, either way probably gonna come out to be the same amount of points, but I might keep it open jaw. Why? Because say your plans change and you don't want to change or miss the domestic flight, its possible the rest of your itinerary will be cancelled.
Either way whether its a bus or a plane, you can always get from rio to sao paolo in a few hours, buying the ticket a week out might be $30, same day might be $50-$100, depending on the airport and time.
Uber is cheap as dirt, esp with the real in the toilet right now. 99 is a good alternative app to uber as well you should download. Public transport I wouldn't bother, you won't even be able to figure it out as a foreigner without porutugese (no offense). Plus public transport can be surprisingly pricy, its almost same to just uber. Also don't forget about all the bike stations peppered around zona sul in rio, you can unlock them using some app and great way to get around as well.
Don't rent a car, you will regret that so bad.