1. Is it necessary to pre purchase a train ticket from Madrid airport to Madrid train station? I see from the Omio app that it is $6 and runs on the hour and takes a 1/2 hour.
What terminal are your sons arriving at at MAD? If arriving at Terminal 4, I would recommend the following advice.
If your sons purchase a Renfe AVE (i.e. the Spanish national railway operator's high-speed train) ticket from Madrid to Seville, that ticket actually includes travel at no extra cost on Madrid's suburban rail network (Cercanías) up to 4 hours prior to the departure of the AVE train.
You simply scan the QR code on the AVE ticket and it will let you through the ticket barriers at the Cercanías ticket at Barajas airport, from which you can take line C-1 to Chamartín and transfer to any train going to Atocha (C-2, C-3, C-4, C-7 or C-8).
If your sons are arriving at T1/T2/T3, the Cercanías station is unfortunately only accessible from T4. In this case, I would recommend taking the express bus that runs from the airport straight to Atocha station.
(A cab is, of course, always a possible option! If your sons are reasonably comfortable on public transport, however, I don't think any of the options from MAD are that difficult.)
2. If they pre purchase a train ticket from Madrid train station to Seville train station for 1:00 pm for example, can they take the 12:40 if they are there in time? I'm wondering if the ticket agent will allow them to take an earlier train if there is space available.
Long-distance rail tickets -- such as from Madrid to Seville -- are for specific trains, like how plane tickets are for a specific flight and they can sell out, since every passenger has an assigned seat. It is best to plan conservatively for the particular train.
3. From the Rick Steves 2017 Spain book (the most recent one I have), it says the Spanish train websites don't accept US credit cards. Is this still the case with US chip credit cards?
I have not had a blanket issue with US credit cards, but will say that I have had better luck with Visa/Mastercard than with American Express.