1. You can spend the whole week alone in Bacelona and still have plenty of things to see and do. Do one day trip to Montserrat if you like but I wouldn't do two day trips. I used GoCity Bacelona 2-3 days pass, it cover most of the Gaudi's museums and include 1 day hop-on-and-off tickets. Most of the places you need to reserve days ahead for the time you want.
2. Madrid, stay in centro. For example, along Gran via and Chueca. You can stay in further north and it easy and cheap to catch a ridesharing car to centro in 10-15 min as well. Centro Madris is small so if you stay in centro you mostly walk from place to place.
3. I will leave others to recommend 'cause I never been to Serville.
4. Low cost airlines are just low cost airlines. Not good but not bad but serves some travelers well. There will be luggage restrictions and you will get charged it you bring a big suitcase. Iberia is decent airline it is not a low budget airline, trying to avoid secments served by LEVEL as that is a low budget air partner which you have to pay for anything extra like a blanket or a drink or a meal.
Soulds like you have already decided to use train from Madrid and Bacelona, good for you. Some of the high speed train ticket includes free local commuter train ride on the day you ride from Bacelona Sants to Madrid Atocha. If that is something you are interested in please google it yourself.
Also, public transpotation is decent but for three people, I would just use Cabify in Bacelona and Madrid, cheaper than Uber and usually 10-15 euro from place to place., not much more than three T-Casual or T-Family tickets.
