1.: yes you can.
2. print-at-home tickets (including TER tickets) do not need to be validated at machines (you only need to validate ATB-style tickets bought at station). Alternatively, you can also buy these tickets via the sncf.com app, in which case you do not need to print anything at all: just show the flashcode in the app to the ticket inspector (and use the flashcode to go through ticket barriers where ticket barriers exist, such as in Nice central station). Although when buying online, you buy a ticket for a specific train, your ticket is, I believe, also valid on other TER trains on the same day. Make sure that you buy a ticket for a TER ticket and not a ticket for a TGV or intercite train as these are not interchangeable. Once bought, TER tickets are not changeable nor refundable.
When buying online, you can buy either through the main sncf website or the regional TER site (
https://www.ter.sncf.com/sud-provence-alpes-cote-d-azur is the one you want for the Nice region).
3. There is a travel agency in Ventimiglia station that will sell SNCF tickets (at the far left as you enter the station). I do not know the times of opening. It might be easier, though, to either buy your ticket online or buy it at a French station in advance (can't remember whether you can buy TER tickets departing from other stations or on different days in automated machines but they should in any event be able to sell you that at the counter of any station).
If there is more than one of you and/or if you have other trains to take beyond those four, you may want to give some thought to buying a "Carte Zou 50-75%." This will give you 50% discount on all TER trains in the PACA region (broadly from Arles in the West to Ventimiglia in the East) for you and up to 3 other persons traveling together with you. There is a form to fill but they can issue the card on the spot at Nice main station and the card is valid for a year. The card costs 30 euros so you need to do the maths to see whether it is worth your while or not. Further details (in French) available
here .
If you have a carte Zou, your tickets can also be loaded on the card (not sure whether it also works for companions or just the card holder) and you need to validate it before each trip by touching the validators at the entrance to the platform.