Seville to Madrid airport

I assume you mean Cádiz? There is a bus and a train, but it’s quite far, so you would need to spend the night before your flight in Madrid.

Our flight departs Madrid at 12:30 pm?

You would be wise to spend the night before in Madrid.

So you have to be at the airport around 9.30 - 10.00. You can rent a car or take a taxi from Cadiz - google maps says it's a 6 hour drive so you'd leave around 2-3 am or you can spend the night Madrid.

Assuming you mean Cadiz, check to see if there is a flight to Madrid from the airport in Jerez de La Frontera. That’s where we caught our very early morning flight back to Madrid back in 2011. Believe the flight departed at 7am and we had plenty of time to make our connecting flight in Madrid.

I wouldn't want to buy a separate airline ticket from Cadiz to Madrid with a transatlantic flight hanging in the balance the same day. If you missed the stand-alone Cadiz-Madrid flight or it was cancelled or substantially delayed, you'd have to buy a replacement ticket from Madrid to your home airport. I'd definitely want to be back in Madrid the night before the flight home.

Trains take between 4 hr. 4 min. and 5 hr. 15 min. just to Madrid's Puerta de Atocha station, then there's extra time required to get to the airport.

Thanks for all of the suggestions. Yes, heading should have been Cazid to Madrid Airport! Seems like the best, and safest, would be to stay at a Madrid airport hotel the night before the flight Madrid to US. That means cutting a day off my Cazid stay which is not good.
May need to find another place to visit for a few days after the tour ends in Seville before heading back to the US. Any suggestions?

First, it's CaDiZ. No one's correcting your topic title, just your rather mangling the name of the city. You could spend 1-2 nights in Cadiz if it's a priority. You can see the sights in one full day. My top choice would be Cordoba which RS skips completely and is one of the best cities to visit in Andalusia.