Using Trains in Spain

Could your daughter ship some of her things home? That would make things much easier. If you google "shipping luggage from Spain," you will see several options.
This page has info on the luggage requirements:
https://www.renfe-sncf.com/rw-en/prepare-your-trip/before-travelling/Pages/luggage.aspx
I would at least check flights to Sevilla, as you may find a slightly higher cost to be worth it in convenience. When I flew into Madrid and took a train to Sevilla, I gave myself something like a few hours.

The general rule for luggage on trains is that as long as you can carry it yourself you can bring it onboard.

There are usually two daily direct trains between Seville and Barcelona so no need to change in Madrid. Buy in advance and you can get pretty cheap train tickets.

While the train is an option, you are right in thinking that it is difficult to time properly vs plane arrival.
You might want to look at a multi-city flight: arrival in Sevilla, departure from wherever your Spanish trip ends.

And cheap domestic flights are no longer cheap once you factor in luggage fees, so factor that in when looking at Sevilla-Barcelona prices.

Otherwise, nobody checks luggage allowance on trains. Just don't bring anything you can't carry or that is too bulky.

I know the current question is how to travel around Spain, but ultimately your daughter needs to figure out what to do with all that stuff when she heads back to the US. The question about how to get the belongings of study-abroad students home comes up rather often here, and there simply is not an inexpensive solution beyond the obvious one of maximizing use of the luggage allowance each of you will have on the plane back to the US. It will help if you and you husband head out with under-packed (lightweight) bags as large as your airline allows; that way, you can each transport some of her things when you head home. Of course, those as-large-as-allowed bags will be a pain to deal with on Spanish trains and will probably cost more on any Spanish flight you decide to take, so that's a trade-off to consider.

In the end, your daughter is going to have to pay to get some of her stuff home, give it away to friends remaining in Spain or discard it. It may turn out that paying extra-bag fees on her US-bound flight will be less expensive than shipping charges. I encourage you to do some Googling to check on those costs, because I strongly suspect that, in the end, she'll decide some things aren't worth the cost to ship or carry on the plane. If she's going to discard them anyway, better to do that before she drags them around Spain with you. Getting even one large bag onto a train, and finding a place to store it, is annoying at best.

What great information. Thank you everyone! I appreciate your time!

After you get through passport control and collect your luggage, follow the signs to the train. You'll have to take a local train to the main station (Atocha) in Madrid to get the Sevilla train. Buy your tickets to Sevilla at the airport. They'll find the next train you can connect to and they'll give you tickets for the local train to Madrid (incl. with the Madrid-Sevilla tickets). It's not worth the stress and possible loss to buy tickets ahead of time.

Train is 5.5 hours. Short taxi ride from your hotel to the station. You can relax, even have a picnic meal. No hassles. At the Barcelona station, you can easily take the metro to wherever you're headed.

Flying means lots of lead time at the airport. Longer (and more expensive!!) taxi to the airport. On landing possible wait to collect luggage. Then Aerobus to Plaça de Catalunya (City Centre), then taxi or metro to wherever. Aside from the extra cost for the taxi in Sevilla, check how much the airline charges for luggage, and look for other fees - like choosing seats or printing boarding passes (you'd be surprised). Keep in mind that cabin baggage limits can be quite low and enforcement (both size and weight) can be strict.

Overall, are you going to save more than an hour or so door to door? Is it worth all the hoops?