Time to allow from landing in Rome to catch train to La Spezia

Getting through immigration is hard to guess even with special passes. I personally wouldn't buy a train ticket ahead of time. Things happen with flights quite often (delays, cancellations, huge customs lines, etc). Also rail tickets in Italy don't offer huge discounts for advance purchases so buying ahead doesn't do too much for you (and if you miss the train you probably have to buy a new ticket anyways). If you want to buy ahead I would try to give yourself about four hours between scheduled landing and a train departure. It is better to kill time than to rush for a train. Otherwise just wait until you are at the train station and buy a ticket for the next train option to La Spezia.

You should make the direct train FB9774 at 13:57, but I too wouldn't reserve in advance.
If you don't make this one, lunch nearby and catch the next direct train two hours later, IC518 at 15:57.

I hate to disappoint you but you will have to stand in line with everyone else. The GOES, commonly called Global Entry, is for ENTERING the US only. The Italians could careless. I would allow a min of three hours to four hours.

Tar,

As Frank mentioned, I don't think a GOES Pass will get you any special treatment in Italy, and you'll have to stand in the "Other Passports" line along with everyone else. As I recall, there are only two lines - E.U. Passports and "Other" Passports. Customs won't be an issue, so you'll be able to walk through.

There's no way to accurately predict how long it will take in April to get from FCO to Roma Termini as there are a lot of variables. At a minimum with perfect conditions it will likely take at least two hours, which includes the 32 minute trip on the Leonardo Express. With that in mind, your best bet is probably a FrecciaBianca that departs Termini at 11:57, arriving La Spezia Centrale at 15:37 (time3H:40M, reservations compulsory). The next direct train (no changes) after that is at 13:57, arriving at 17:13. From La Spezia you'll need to take a Regionale to whichever of the five towns you'll be staying in, which only takes a few minutes.

You can buy tickets for both the Leonardo Express and your other trains right at the FCO rail station, either from a Kiosk or staffed ticket office.

While you could take a chance and buy advance tickets for the Freccia train, that's not usually recommended by most here. If there are any delays in your flights and you miss the train you've booked, the tickets will be worthless and you'll have to buy more at full price.

Have to disagree that advance purchase doesn't save much - a super economy ticket from Milano Centrale to Venezia Santa Lucia can be had for €9 if bought up to 120 days ahead. All fast trains have super economy fares (until those tickets sell out), regionale trains are generally not discounted.