Travelling from the Madrid airport to hotel in Córdoba via the Renfe train

1. It's not easy to say. Madrid Barajas airport is absolutely massive, especially if you're landing at Terminal 4. If you're landing in the morning from a redeye from America, it won't be crowded in most of the airport (passport control probably won't take long), but it can just be a hike to walk all the way to transit. Obviously, you also have to consider that your flight could be delayed. You could book a flexible train fare where you can change for free if you miss your train (I know the spanish train company Iryo offers this).
2. I've used the express airport bus and it's not blazing fast, but it works. It took us about 40-45 minutes from Atocha to T4. You could also get a taxi, cabify/uber, or take the commuter rail. (if you're taking an AVE/Renfe train to Cordoba, commuter rail is free for your trip to the train station, just scan your Renfe ticket). I've had very good experiences with the Cabify App in Spanish cities where it's available, that might be the fastest, and they have a nice guide for getting their taxis at the airport.
3. Not necessary. You pay with contactless credit card or Apple Pay on the bus. If you've used the London tube contactless scheme, it's the same thing as that.
4. I don't see why not. There's a big city park outside the station that runs north/south, you can walk down there in the shade until you're next to the historic center (I assume you're staying there). If you're very tired from traveling that much, get a taxi. There will be a taxi stand outside the train station.

@plectrude Thank you very much for taking the time to share this information, it's very helpful! We really appreciate it :)