My perfect weekend in Madrid would include lots of time spent at the Prado over at least two visits, plus time at the Thyssen, and time at the Reina Sofia, with walks in the Retiro Park and the Botanic Gardens in between. That's two of your four interests covered, and the others can be satisfied by just strolling around the central area to see what you can se of this lively, interesting city.
There are a number of other excellent museums, but those are the three major ones, and I doubt you'd have time for any more. If you do, the Real Academia de Bellas Artes de San Fernando has a fine collection of Spanish painting, including a large number of Goyas.
Have a look at the Hostal Gonzalo. I've always stayed there, and love it. It's quiet, comfortable, clean, reasonably priced, run by a friendly, helpful family - and a few minutes walk from the Prado and the Thyssen.
Edited: 3:50 am, November 22, 2012