Open Train Ticket?

You didn't find Cinque Terre because it's not a city, it's five towns, each with its own station and name.

The rail.eu website probably has a contact number or a Frequently Asked Questions menu, or perhaps there's an explanation that came with the tickets?

Those two train routes purchased from rail.eu don't usually require seat reservations. The Swiss leg should indeed be an open ticket, valid any time during the printed validity dates.

Italian Regional train tickets used to be like that, but recently added a 4-hour window of use for tickets purchased online; seat reservations not available. If the ticket is for a regional train, then don't get onto a reserved InterCity train.

I'm more concerned about your ticket departing Arth-Goldau. Is this a print-at-home or "ticketless" ticket? Over the past few years, Trenitalia required that international tickets be picked up in an Italian train station.

Courtney, you don't reserve seats on Swiss trains, tickets are valid on any train, just get on. There is usually no discount for advance purchase so there is no need to buy in advance, it is the same price if you buy at the station on the day.
Times and prices can be seen on the Swiss Railways site: https://www.sbb.ch/en/

Trenitalia is the Italian national rail company. They only sell tickets for train within, to or from Italy, that is why they cannot sell you a ticket for a train Luzern to Arth-Goldau.
I agree with Laura's worry about the ticket from Arth-Goldau. I hope it isn't "pick up at any Italian station".

As the others said Cinque Terre translates as "five lands". It is five towns, you have to decide which of the 5 you want to go to and buy a ticket to that town. You said you purchased a ticket, but to where? What does it say on the ticket?

Who do you mean by "rail.eu"? I went to http://rail.eu/ and the internet address is for sale.

In terms of Lucerne to Arth-Goldau, it sounds like I'm set as there is indeed open seating on the train. RailEurope mailed me paper tickets.

In terms of Cinque Terre, I purchased a ticket from Florence SMN station to La Spezia, with a connection in Pisa and then will take the regional train from La Spezia to Corniglia (planning to get those tickets at the station). I have a paper ticket that gives a range of dates of validity but not a specific time. I was mainly just concerned that that ticket did not have a specific train time on it. Sounds like I'll be able to use it on any regional train though?

Courtney, In Italy tickets for Regionalle trains are "open". But you must stamp the ticket in a machine on the platform before getting on the train. This is called "validation" and is to stop people using the ticket twice.
If the ticket lists a specific train on it, then you don't need to stamp it as you can't use it twice.

For lots of info about Italian trains see here: https://seat61.com/Italy-trains.htm About 2/3 of the way down there is a photo of a validation machine.

Switzerland doesn't use this system, no stamping needed, your ticket will be checked on the train.