Train from Cinque to France?

With a 2 or 3 changes, in 4 1/2 hours or so, depending on where in the Cinque Terre you start from.

Sure. If the town has train service you can get there via train. May require a number of changes depending on where the town and type of service. If asking for a direct train - probably not just like airline service.

Yes. It will take 4-1/2 to 6 hours, depending on choice of departure (according to the Deutsche Bahn website). I think all will require transfers at Genoa and at Ventimiglia (border). Some may mean a third change at La Spezia.

You can check schedules on the TrenItalia website but I think you may only be able to buy tickets there as far as Ventimiglia. Examine trip time carefully, because you can pay quite a bit extra for a trip that includes a leg on a $$$ fast train but takes only a bit less time than some of the considerably cheaper all-regionale itineraries. Edited to add: All-regionale trips do not need to be pre-purchased; you should not buy those from the US. Retain your flexibility!

The French railway will price the full trip, but I see that the shortest travel times are not there. If you try scheduling the two pieces separately (CT to Ventimiglia, then Ventimiglia to Menton) you may be able to figure out something. Menton is extremely close to Ventimiglia, so I don't think there will be a huge price penalty if you just plan to buy the second, short, leg at the Ventimiglia train station. I hope someone else knows.

I did this last July from Monterosso al Mare with a change in Genoa. It took 5 hours
you can buy from this service
https://www.thello.com/?iLangID=3

There are frequent trains from Cinque Terre to Genoa (around 2 hours, depending on which village you're coming from).

From Genoa, you can take a train to Ventimiglia (about 2 hours 20 minutes for the regionale train or 2 hours for the fast trains).

In Ventimiglia, change to a train to Menton (about 15 minutes).

The tickets from Cinque Terre to Genoa and on to Ventimiglia can be bought in Italy, but the train from Ventimiglia to Menton will need to be bought at Ventimiglia station. It's only €3.20 and runs frequently (about every 30 minutes during the day), so no need to worry about buying it in advance.

If you take the regionale trains, wait and buy your tickets once you're there - the price won't increase and you'll have the freedom to take whatever train suits you.

While you are in the Ventimiglia station getting your ticket you can enjoy some Socca. Although you are in Italy it is made similarly to that in Nice - not as good but grab Socca when you can is my motto.

While you are in the Ventimiglia station getting your ticket you can enjoy some Socca. Although you are in Italy it is made similarly to that in Nice - not as good but grab Socca when you can is my motto.

Yum! Farinata/socca is a great reason to have a stopover in every town along the way - each city and village has their own recipe.

No single railway sells all the tickets, including Thello. Thello operates from Genoa to Ventimiglia or Nice, and also sells Trenitalia tickets between the same destinations, but doesn't serve or sell connecting tickets for either Monterosso or Menton.

Thanks everyone. I think I got all the info I need including a little extra I wasn't counting on. Socca, Yum.