October is kind of all over the place, weather-wise. The last few years were really warm, like 30C highs on some days. But usually you're looking at temperatures more in the 15-25C range. You'd probably want to pack for both summer and fall...
Your flight is scheduled to arrive at 14:35, that doesn't mean that it will arrive then. I would schedule a later train to avoid the stress of rushing to get to the station in two hours.
Hi @beano5892 and welcome. I hope you will get an answer. Many people on this forum (I think most people) don't use tour companies and just manage Camino planning on their own, as it saves money and really isn't that difficult. Best wishes!
The beginning of March is definitely not too early, you could start even sooner. CaminoWeather says that's more or less the sweet spot: https://caminoweather.com/via_de_la_plata/best_time_of_year. Though when I walked it around then last year...
Just for the record, I'll add that there's a Madrid -> Burgos train, which is what I once used. Though to be clear: looking at what others have said about the bus option, I'm not actually suggesting this as a better option.
I'll just add that I've seen Atocha's security lines get backed up in the past, because of construction work. I don't whether they're done expanding it - but unless someone confirms that the train station is functioning normally now, I'd give...
Hostal Equis in Cerro Muriano. https://www.gronze.com/andalucia/cordoba/cerro-muriano/hostal-equis
The owner is dedicated to the Mozarabe, and has a special "pilgrims' room" that's completely plastered with all the Camino paraphernalia he could...
This site is run by Ivar at in Santiago de Compostela.
This site participates in the Amazon Affiliate program, designed to provide a means for Ivar to earn fees by linking to Amazon