flying internationally, then adding another flight later the same day with a local airline

Lets take this a step at a time. Using two unrelated tickets is risky and generally not advised. You cannot check luggage through to the end unless there is a relationship between the two airlines. And generally there is not a relationship with discount airlines. As stated above, you would claim your luggage in Madrid, go through immigrations and customs, and then recheck your luggage with the other airline. The important question is how early can you check in for the second airline? (Don't understand your reference to storage at the Madrid train station - that makes no sense as you are no where near the train station.) The big risk you take is in case something happens to your original flight that causes you to miss the second flight. In that case that is a dead ticket. Granted 10 hours is big time window so you should be OK but there is still some risk. Booking a 4 pm flight adds more risk in case your plane is late.

If you had booked directly to Seville with one ticket then your luggage would go through to Seville, you do immigrations in Madrid and customs in Seville. That is easier and cannot be that much more expensive. Another option would be to taken the bus to the train station, store your luggage, kill a few hours in Madrid, and then take the train to Seville. And hope that someone wakes you up so you don't miss the Seville stop.

PS This posting is a little confusing in light of your other questions. What exactly are travel arrangements?

Since these discussions have educational value for travelers other than the OP, I am adding a slightly unrelated comment. Taking an international flight and then a discount airline to another European local is frequently discussed and recommended by some - not me. Using travel dates of mid May 2015, I found a difference in fare between flying into Madrid or Seville between $100 and $150. Maybe you can get a discount airline fare to Seville for less than $150 but discount airline fares can be tricky with all the extra fees and then there is always a risk with two separate tickets. IMO, this is really a false economy to save maybe fifty bucks. I can save that $50 quicker with a better selection of a hotel room and not have the hassle of a late connection or extra fees.

Why not just take the train?
A 2.5 hr. non-stop AVE leaves from Puerta de Atocha every half hour.
With smooth going at the airport through customs and passport control,
you could be in Sevilla's Santa Justa at 4.

@mph,

"Even if you were making the entire trip on one ticket, which you are not, you would have to collect your bags and pass through customs and passport control as Madrid is your entry point to Spain."

That hasn't been my experience after many flights to Europe. I always use code share flights (one ticket) and my usual point-of-entry to the E.U. is Frankfurt, where I go through Passport control. I do not collect my checked luggage there. In every case, I check my Backpack at my home airport and don't see it again until I reach my final destination.

Unfortunately the same is not true on the trip home, as I always have to go through the ridiculous ritual of collecting my checked luggage at my point-of-entry here, walking it a few steps and placing it on the conveyor belt for the final flight to my home airport.

saw,

You likely won't be able to see much walking around Madrid for just a few hours, but if that's important to you it's certainly possible. Given the circumstances you mentioned, I wouldn't bother booking another flight as the rides back and forth to the airport would use up some of your valuable times. In this case, travel by train would be a better option (IMO). There are direct high speed AVE trains on that route with a travel time of about 2.5 hours, which is probably going to be much faster than by air, when all is considered. The trains travel at up to 300 kmH so very quick.

After leaving the airport, you could go directly to Madrid-Puerta de Atocha station. According to THIS website there are luggage lockers there behind the Aviary. Spend a few hours looking around Madrid and then return to the station for your train to Sevilla. I'd suggest not leaving it too late, as I never find it pleasant when I arrive late in a new city. Also, with earlier travel you'll have some time to admire the scenery during daylight hours. A departure from Madrid at 19:00 is about the latest one I would take. I can't remember the stations in Sevilla, but your destination will probably be Sevilla Santa Justa. You can see all the details on the bahn.de website.

If you decide to use this method, it would probably be prudent to pre-purchase your rail tickets. However, one point to note is that they will have compulsory reservations so will be specific for one train, date and departure time. Don't be late getting back to the station after touring Madrid or the tickets will be worthless!

If it is too late to book a code-shared Seville flight connection through United, then why not reduce connection stress and stay overnight in Madrid?