The trains that normally don't require a connection at Ventimiglia are operated by Thello, which says that those departures are temporarily closed for sale for travel on July 3, 4, and 5. I would guess that the issue is construction, rather than a strike, since the alternative of "regular" Italian regional include a bus for the portion of the trip between Taggia and Ventimiglia. But then you still have minimal train service on the French side, too.
As I think you saw, SNCF and Eurolines are both selling a direct Eurolines bus that would be the best bet for advance booking. The bus stop at Piazza della Vittoria is about three blocks from Genoa/Genova Brignole train station.
12:15 Genoa - Piazza della Vittoria
15:45 Nice - Aéroport Côte d'Azur - Terminal 2
There are train connections about hourly from Monterosso to Genoa, most with one connection. Looking Up Train Schedules and Routes Online gives you the Deutsche Bahn train schedule link and tips for using it, or those would also be on www.trenitalia.com.
