Another website to check for apartments in Madrid is homelidays.com. You probably won't find many in the Sol area, and I agree it's too noisy. As mentioned, the Plaza d'Espana area is nice, and within walking distance....a couple of other locales include Chueca and Fuencarral.
As for the southern portion of your trip, I would take the train. It's a 6-hour drive (before stops), and you'll find that renting a vehicle in Madrid large enough for 6 people and their luggage, and dropping it in Malaga, will probably cost you close to 100 euro/day. If you book early (up to 62 1/2 days in advance,) you can get train tickets for as low as 35 Euro per person, and it's less than 3 hours. You can then jump on the local commuter train (the cercanias, included on your RENFE ticket) out to Malaga airport (about 15 minutes) and get a similar vehicle for far less....about 70 Euro/day, and the CDW is zero.
As for your return journey, there is no overnight train from Malaga to Madrid. So your options would be to take the last train in the evening then hang around Barajas airport for a couple of hours before your flight check-in begins around 4 a.m.....or take an earlier train and getting a hotel so you can get at least a few hours' sleep. With a long flight home in front of me, I'd choose the latter.