How to purchase trenitalia frecciargento tickets from US

Nope. I don't think they are mailing anything. Just use the address of your hotel. A quirk of the system. They are not sending anything so the mailing address is unimportant.

Another alternative is Italo Train, http://www.italotreno.it/EN/Pages/default.aspx, I will be using them next month for my Rome - Florence - Venice portion of my trip. Their site offers an English version, easy to use and has discounted prices when purchased early. Able to choose seating before checking out and offers 3 levels of seating type. You print your tickets yourself.

Trenitalia.com can be translated into English by clicking on the British flag icon near the top of the page. Your hotel's name will work as an address, you will receive an e-ticket within seconds of completing your purchase. You can either print out the ticket or show the conductor the screen ticket. I prefer carrying a paper copy.

You can purchase high speed train tickets as a guest without logging in or creating a id. They will email you the ticket right after purchase for you to print out

Thank you! I appreciate all the advice.

How risky is it to purchase a trenitalia ticket from Florence to Venice, when we will only have 25 minutes to change trains (coming into Florence on a Regionale train from La Specia)?

I would be fine with it. Florence is not that big a station.

FYI: Italo trains now depart/arrive at Rome's Termini station but only twice a day. More frequent service is available through the other two Rome stations.
http://www.italotreno.it/SiteCollectionDocuments/italo_orari.pdf