Just wanted thank everyone for the fantastic info. My mission to Monaco went off without a hitch. It IS possible to leave Chicago at 4pm Saturday afternoon and be in your seat at Monaco two hours before the start of the race (I wouldn’t recommend such methods, but amazingly it worked).
I ended up taking Bus #23 from the Nice Aeroport to the St. Augustin train station. Only about a 3 minute ride, then a 3 minute walk up the hill to the station. Train was packed by 10-1030am, really had to squeeze in. There was no more room on the train by the third stop after St. Augustin.
After the race, the funneling of the crowd around the “back” entry of the Monaco train station was a bit chaotic. Waves upon waves would get held up by police as they let waves into the station slowly and orderly. If you have to leave in a hurry after the race and are a ways from the station, you may have to miss the last few laps in order to make your train.
The other two trains, from Ventimiglia and then Genova to Sestri Levante were not crowded. Plenty of room.