culpeps,
Regarding buying tickets in advance, I wouldn't bother for that route. The trip from Nice Ville to Ventimiglia is short and I doubt that you'd save much money with pre-purchased tickets. Also, if the particular train you were using required compulsory reservations, you'd be locked into travelling at a specific time.
For the trip from Ventimiglia to the Cinque Terre, I had a brief look at the Trenitalia website. Using an arbitrary destination of Monterosso, it appears that many of the trains on that route are Regionale. It's difficult to buy tickets for those in advance and absolutely no price benefit for doing so. It would just as easy to buy the tickets at Ventimiglia, either from a Kiosk or the staffed ticket office (which may have a queue). Of course, this may vary depending on which train you connect with in Ventimiglia. You could also be using an Intercity on one leg, which requires compulsory reservations, but those will be provided with the ticket.
One important point to mention when travelling on Regionale trains. You MUST validate (time & date stamp) the ticket prior to boarding the train on the day of travel. Those found with unvalidated tickets are subject to hefty fines, which will be collected on the spot. The tickets have a "shelf life" once validated (about 6 hours, as I recall but that varies with the route), so don't validate until you're ready to board the train. That doesn't apply to trains with compulsory reservations, since the reservations are specific for a particular train, date and departure time. With tickets that require compulsory reservations, you MUST board ONLY the train specified on your ticket, or again hefty fines.
Regarding accommodations in Monterosso, I have no idea whether there are any AirBnB properties there as I always stay at Hotel Villa Steno (which is one of my favourites in all of Europe). Hopefully one of the others can provide information on AirBnB in that area. One other point to mention is that Monterosso is the only one of the five towns that has "conventional" hotels. Accommodations in the other towns consist of small rooms or apartments, some of which don't provide breakfasts (I like a nice breakfast, so that's important for me). I'm sure one of our regular posters will be along shortly to provide a recommendation for THIS facility in Vernazza.
Cheers!