Sufficient lead-time to catch a train at the Madrid Atocha station

Yes, but you hit the key phrase - assuming the flight is on time.

You don't need to take a taxi (although you can if you wish). You can take a cercanias train right from the airport to Atocha, and the price of this train will be included in your Madrid to Barcelona ticket. Just ask when you buy it, how to activate the cercanias code. Alternately, if you don't use the code getting from the airport to Madrid Atocha, you can use in Barcelona to get from Sants station to your hotel. Note that in Barcelona, it's called the rodalies instead of the cercanias. I found the Sants station personnel very helpful with getting my rodalies code to work, and which train to take to my hotel.

How much of a gambler are you? Right now you can buy 2 tickets for €150.80. The maximum price you'll pay on the day is €215.40. If you buy now, you save about €65, but you could lose €150.80 if you miss the train.

Harold, Thank you so much for all the useful info! There are 4 of us traveling (my wife and I plus 2 teenage sons), I plan to purchase the tickets through the renfe.com website, they have the "Tarifa 4 Mesa" deal for the Turista class for about 43 euro/person oneway. But I do not see the Rodalies code option during the online purchase process, though. We will need to use the Barcelona metro to get to our hotel. On a separate note, I read that Cabify service is very much similar to Uber in the States, and it's both cost and time efficient to use it (for 4 adults and 3 mid-size luggages) between Madrid airport and Madrid Atocha. Do you have any experience in that particular way of travel? Thanks again!!

Chani, thank you for your advise. There will be 4 adults in our group (adults tickets required at least), I plan to purchase the tickets through the renfe.com website, the "Tarifa 4 Mesa" deal for the Turista class for about €43 euro/person oneway. So about €172 total. So about €258 of savings if we can make it to the train in less than 3 hours. I know it's still a gamble, that's why this is such a tough decision to make.

"I do not see the Rodalies code option during the online purchase process, though. We will need to use the Barcelona metro to get to our hotel. "

You will not see the rodalies or cercanias code during the purchase. But when get your ticket (it's emailed to you as a PDF), on it is a six digit combination of numbers and letters, labeled "CombinadoCercanias" (I just checked my old ticket). You can enter this into a machine to get a rodalies ticket. However, note this is only good on the rodalies or cercanias trains (commuter rail), not on the Metro, which is a different system.

If you say where your Barcelona hotel is, people can tell you which rodalies line to take.

"On a separate note, I read that Cabify service is very much similar to Uber in the States, and it's both cost and time efficient to use it (for 4 adults and 3 mid-size luggages) between Madrid airport and Madrid Atocha. Do you have any experience in that particular way of travel?"

No. But from the airport to Atocha is a fixed rate of €30 for up to four passengers, and no tipping is required beyond this. I took a taxi in from the airport, even though I was one person, and felt it was money well-spent, in my jet-lagged state to avoid having to find an unfamiliar hotel address and schlep bags. So, I agree with the idea of a taxi, particularly for 4 people. In fact, you may want to take a taxi from Barcelona Sants to your hotel, as well, for convenience, particularly since the cost will be divided by four.

In Madrid and Barcelona it is better to use taxis or public transport. They are cheap and super reliable. All customers on taxis are protected by the Regional Government, and the Regional Government sets the prices. And there are lots of taxis always in Madrid and Barcelona ready for you anywhere, so no need to prebook nothing. Alternatively, take Cercanías C1 to Atocha Cercanías free of charge with your CombinadoCercanías code, but they run every 30 minutes, so if you are in a hurry, take a taxi.
A taxi is a good option to go to Puerta de Atocha (Departures). Once in Atocha, you should go to the upper hall security control, because from there run the High Speed services to Barcelona. Pass the security control (brief, only takes 30 seconds and it is nothing compared to an airport), then you wait in the boarding hall until your track is assigned.
At Sants, go to any orange machine that has Rodalies de Catalunya signs. Touch Combinat Rodalies and scan the QR codes of your ticket at the barcode scanner. Alternatively, type the CombinadoCercanías code manually.
Once you all have your tickets: if you go to Plaça Catalunya, take any train on Platform 8. If you go to Passeig de Gràcia, take any train on Platforms 13 and 14. If you go to Estació de França. take a train on Platforms 13 and 14 towards Barcelona Estació de França.
All Renfe tickets for Long Distance services (AVE, Alvia, InterCity, Euromed...) include the CombinadoCercanías code by defect.

Hi Joan, thank you so much for your input and all the useful information you've provided. This is extremely helpful. Thank you so much! God Bless!