An antibody test should be very reliable. I believe these are coming though making them available to 330 million people is going to be a big challenge.
I doubt there would be temp checks for games played in February and beyond. That doesn't seem very practical to do, especially outside in the middle of winter. It is becoming clear there are a lot more people who have it that are asymptomatic and would be let in anyway. You would be inconveniencing many for what would amount to a minor benefit.
Let's assume for the moment that it is formally proven that once you get the virus, you can't get it again, at least for a year or two. There may or may not be a vaccine by then. Even if there is, it is likely not everyone will have been immunized.
Given all this, how could athletic events be staged with fans in a practical way? There is talk about people with immunity getting some kind of proof from the government that they are immune. That might be something that happens, especially if a vaccine is not generally available for a long time. That proof might be leveraged in these situations. It is a dicey subject giving special treatment to those with immunity but it might be something that is needed to keep the economy moving and provide enough support to allow people without immunity to stay safe.
If we do end up playing football in the late winter/early spring, I would expect Syracuse to play a lot of home games early in the season. It would not surprise me to see the dome hosting football games for other colleges on weird days and at weird times as well. Schools like UB, Cornell, Colgate, Ithaca and Albany might have no other alternative to getting their home games in early in the season.
This would be a time when being in a conference with a lot of schools based in the South helps a lot. I don't know what the B1G in particular is going to do.