8 a.m. is a busy time for arrivals in T3, but FCO is a surprisingly efficient airport all things considered, so there shouldn’t be too many delays getting out of the airport, except that your luggage may take some time to appear on the carousel. If avoiding long lines upon arrival is your only motive for pre-booking your tickets, then one way to avoid that is to bypass the ticket machines in the Arrivals Hall and head straight to the railway station, where there is generally a longer line at the ticket counter, but I have rarely seen a queue more than two persons deep at the ticket machines, if you feel confident enough to buy them that way. Given that you’re arriving on a Sunday, it may nevertheless be prudent to book in advance. The best way is of course through the Trenitalia website, as you suggest.

In the event of a delay, it is the second FrecciaRossa leg that is more critical. Unless the discounted fares (SUPER-/ECONOMY) are amazingly cheaper, I would recommend your purchasing the BASE fare, to give yourself more flexibility. Note though, that, even so, you only have an hour’s grace, i.e., you can change the ticket to a later train up to an hour after the original departure time. (This can be done at the Fiumicino station; you don’t have to wait until you get to Termini.) If your departure from the US is severely delayed, however, you should do so online before you take off, as you might not make it to the Airport Station before the grace period elapses.