Wow! I am impressed at the specificity of your questions!
1. You can certainly purchase a ticket in advance on the Trenitalia website, using a US credit card. You'll get a confirmation with the reservation PNR number. This is the key to the reservation. You need to present this number to the ticket agent on the train when the ticket is checked. The very best evidence is to print out the entire e-mail you receive from Trenitalia, as it contains other information which may become necessary if, for example, you needed to make changes to your reservation (which does not apply in the case of the Super Economy fare). However, you could also write the PNR number on a paper napkin or on the palm of your hand or on a scrap of paper ......
2. I have a characteristically "Italian" attitude towards advance purchases of train tickets for the unreserved regionale trains: because all seats are unreserved and the trains never sell out, and because there are never any price discounts by purchasing in advance, I prefer not to shell out any money until the very last moment, as that preserves total flexibility for a multitude of contingencies and unexpected events, both known and unknown. I make an exception if I am CONNECTING TO an unreserved regionale train from another train, and the connection time is insufficient to allow me to safely buy the ticket for the second train at the connecting train station without risking a missed connection. However, you might have enough time to buy this ticket at your originating train station (before boarding your first train). If so, you must enter the departure city as well as the arrival city on the keyboard of the electronic ticket machine when buying the paper ticket. The departure city will be different from that programmed into the electronic ticket machine (since you are buying the ticket in a different city from that where you will be boarding the regionale train).
3. In case of a strike which actually affects your travel itinerary, you would be "protected" in the sense that all of your tickets would be honored for substitute travel later in the day, or on the next day. However, you would need to get your reservation for the reserved train changed. A PAPER ticket for the regionale train can be used on ANY regionale train departing before midnight of your stated travel date. Beyond midnight, you would need to get your ticket replaced by a new ticket. BTW, an ELECTRONIC ticket for a regionale train is valid for only four hours after the stated departure time of the train noted on the receipt. Beyond the four hours, the ticket expires, so in case of a strike, you would likely need to get the electronic ticket replaced.
4. Registering is not necessary to buy electronic tickets on the Trenitalia website. Registering is helpful if you want or need to manage your account online (such as for future changes or cancellations). For the Super Economy ticket, this is a moot issue.
5. You should be able to buy an electronic ticket on the Trenitalia website using a US credit or debit card, especially if you notify your bank in advance.
6. US debit and credit cards with a PIN can be used in Trenitalia electronic ticket machines. If you have only a magnetic strip, you would need to stand in line at the manned ticket sales counters. Personally, I avoid using plastic in the machines unless I've run out of cash and there is no ATM nearby which I can use. This is because I am rather fussy about avoiding potential frauds. I have, however, used plastic in the machines on occasion in the past, with no problem.
Hope this helps.