Hello AnotherAtlantaMike! Welcome to the ALL SPAIN Message Board! I'd be happy to give you my opinion on your situation. And really, I think it's a pretty easy answer to/for me.
Me, being a 36-YO SWM with a "ladyfriend" IN Madrid, EVERYTHING in central Madrid is within walking distance. I suppose if you wanted to save 5-minutes walking then find a hostel (called "hostal" in Spanish) along Gran Vía as it is about 5-minutes closer to Chueca than is Puerta Del Sol and maybe 10-minutes closer than Plaza Mayor.. maybe even less if you walk swiftly.
I can personally recommend Antonio's
Hostal Chelo ; and NOT because he's an administrator on this message board but because I have spent a comfortable night there and visited him a number of other times during the day. It's located on Calle de Hortaleza, about a block north of the Gran Vía and 2-3 blocks south of Chueca. I don't think they have safes available in the room though.
Not many hostels have safes in rooms, but some do. I recall that
Hostal Cervelo on Calle de Atocha &
Hostal Aresol on the Puerta del Sol did.
There are many many hostels from which to choose in the Plaza Mayor, Puerta del Sol, and Gran Vía areas.. all are a short walk to Chueca and also the other hot spots of Madrid. The Huertas area might be just a little too far to walk after a long night of dancing/partying in the Chueca area, though, but it's also in the heart of the another HOT nightlight area in Madrid (in Huertas).
And yes, all outdoor cafés will be closed by, probably, the end of October. Weather will be cold at night, but Madrid rarely gets below freezing (32ºF/0ºC) in the heart of winter. I was in Madrid over Christmas/New Years and while it did snow one day, I was perfectly comfortable with a leather jacket and liner and not wearing a hat. You can see Madrid's month-by-month climate guide
HERE on the USATODAY.com weather page for that city.
Have a GREAT time! Saludos, MadridMan