By midcentury there would be more disciplines, like rally racing and drag racing, in addition to the basic "Auto Racing" events, which I imagine would probably be Formula One style. Motorcycle racing, too. Well, it would have started with some form of race car in 1900 and it would have evolved over time. It would probably end up emphasizing the fastest possible cars, like F1, but perhaps Olympic Formula cars wouldn't look like modern F1 cars do, because the development history would be different.
I suspect the host country would provide all vehicles. (Who supplies luges and bobsleds for the Winter Olympics? I think it's the host.) For that reason, Olympic performance would be just a shade below standard F1 performance. World records would be better than Olympic records.
I like the idea of kicking off the events with a Grand Prix race, although in practice a few events occur before the opening ceremonies, but I think the main event would be held towards the end of the Olympics, perhaps on the same day as the marathon.
Not sure what you meant by what teams participated in Rio, but here's a medals table for auto racing 1900-2016 (thirty olympiads).
Country: Gold/Silver/Bronze
USA: 61/55/47
UK: 42/53/46
Italy: 6/12/31
France: 12/21/31
BRD/Germany excluding DDR: 39/24/31
USSR/United/Russia: 6/8/24
DDR: 1/4/19
Japan: 2/9/17
Canada: 2/12/21
Rio results:
24 hour endurance: USA/France/UK
500 km hard track: USA/Germany/Italy
250 km urban track: France/USA/Russia
500 meter drag racing: USA/Japan/UK
Motocross, acrobatic [scored like gymnastics]:Canada/USA/Ireland
Motocross, racing: USA/Slovakia/Germany
[The numbers don't add up or make sense, they just give an overview of how I think things would have happened... remember there are multiple events per year. The Rio results are meant to show the events. There are six, about the same number as equestrian events.]