Psyche,
Thanks for the additional information on the tickets. That's not a method I ever use with Regionale tickets, but it should work fine.
I've found that while trains can be late at times, I've never had one "break". The route you're travelling from Florence to Riomaggiore is relatively short and fast, and I wouldn't anticipate any major problems. However, there can certainly be delays with any trains, including the high speed trains. For example, last year I experienced some delays due to some kind of issues on the high speed tracks between Rome and Florence, which affected trains north of there also. Both Freccia and Italo trains were delayed as a result. Especially in Italy, one has to be prepared for things like that (although thankfully they don't seem to happen often).
If there was a delay and you missed the Freccia train that you booked, I'd suggest speaking with a Conductor or someone at the ticket office. That's a situation that you'll have to "play by ear". The segment from La Spezia to Riomaggiore is only about 9 minutes, so that will be the easiest part of the journey.
Have you been to Riomaggiore before? From the station plaza, you'll reach the main part of town via a short tunnel that parallels the tracks. Depending on where your accommodations are located, you'll probably have to turn left at the end of the tunnel and walk up the main street.
Sorry, I missed your earlier question on the TGV. If you have to pick up your ticket in Paris at a Kiosk, I believe that will have to be validated prior to boarding the train. If you're using a ticket which was printed at home, I don't believe those have to be validated. The website link you provided was dated 2008, but is probably still accurate. Have a look at the excellent Man In Seat 61 website, and scroll down to the Collecting Your Ticket part and the section on Ticket Validation......
http://www.seat61.com/France-trains.htm#Travel%20tips
The paragraph after that states....
"All French train tickets (except print-at-home e-tickets) must be validated just before you board your train, by putting them into the small yellow machines marked Compostez votre billet at the entrance to each platform, see the photo on the right. There may be a fine if you don't!"
Hope this helps.