two questions for Italy trip

With a 9:45 am international flight I would not be coming in from Orvieto that morning...I would suggest you arrive in Rome the day before.

Right now, there is a train at 4:30 am that gets to FCO at 6:37 am after a train change in Rome. That should be OK.

Yes, the trains from La Spezia to Vernazza are local and don't take reservations.

Since there is a train schedule adjustment coming up in June, the local trains and many IC trains are not scheduled yet. They usually don't change much from the current, but keep checking. www.trenitalia.com

Valerie is giving you very good advice. Italian rail strikes are common.
With a morning flight I would spend the night at a hotel near the airport.
And you do mean FCO for the main Rome airport, yes?
Happy trails!

Please realize that you should be at the airport at least two and, preferably, three hours before your flight to check in, go through security and get to your gate. Check on your airline's website to see recommended times for your international flights to and from Europe.

Also, you must take a train from FCO to Rome Termini to make your connection to La Spezia. There are no direct trains from FCO and no reserved seating to Rome - this is like a commuter line.

I love Orvieto but you will also have to figure in some hassle time just to get down from the town to the train station via funicular. You might need to plan to leave your lodging at 345 or so??

https://en.m.wikipedia.org/wiki/Orvieto_Funicular

Definitely spend the last night in Rome.

Sam's suggestion leaves you 15 minutes to transfer trains in Rome. It's cutting it a little close, since [1] it's a regional train so delays may occur and [2] the Leonardo Express (to FCO) is usually a bit far from the other trains and you won't be familiar with the station. However, if you miss it, the next one is half an hour later, so you'll get to the airport shortly after 7, which is probably enough time. You will have to take a taxi from your hotel in Orvieto to the train station at that time of night. Whether you stay in Orvieto or Rome on your last night depends on your tolerance for close connections.

I would not buy train tickets in advance for the day of arrival. There are too many variables. If your flight is delayed for any reason, if you have a long wait for your luggage, if there's a long line at passport control . . . you really have no way of knowing how long it will take you to get to the Leonardo Express. Just buy tickets at the Rome station (Termini) for the next train when you get there.

In the meantime, play around on the Trenitalia website. If you are taking regionale trains, they only show up a week in advance, so choose the upcoming day of the week that you plan to travel.

Whew!! That was close! So glad I asked and you all informed me. I will DEFINITELY stay an extra night, then, at an airport hotel! I do not like living dangerously! And thank you for the train advice as well. So I do not need to book trains in advance from La Spezia to Orvieto either?

Right now, there are some all-Regionale itineraries for 26.30 EUR, taking about 5 hours. There are some itineraries using IC trains taking a bit more than 4 hours, and costing less when bought as Economy or SuperEconomy, but they are not scheduled yet. Keep checking.

Great!

Now I will just suggest you plan to spend some time looking at the carvings on the front of the cathedral in Orvieto. They are just amazing. I particularly like the illustration of Eve coming out of Adam's side. This picture is good but it is even better up close!

https://en.wikipedia.org/wiki/Orvieto_Cathedral

I also used Rick's guidebook for the San Brizio Chapel inside the Cathedral. I believe this is a Cathedral that has a small fee to enter. That does not bother me as they have so much extra expense from tourists that I am happy to help support in a small way.

Yes, there is a fee for the chapel in the Orvieto cathedral. Orvieto has a city card that covers lots of local tourist sites and the walking tour organized by the tourist office. It's probably not worthwhile for most visitors, but anyone staying at least two days might want to check it out to see whether it would save some money. Orvieto (or at least the old part, up on the hill) is not a large city, but it has quite a lot of pay-to-enter sights.

If you go to the museum across the street from the front of the Duomo, you can see the facade mosaics much better from the museum's upper floors.