How do you close a Sash Weight Window?
Your windows are very old and things have worn out, warped, etc.  The sash weights are there to counterbalance the weight of the window. When the tracks (guides on each side of the frame) are clear and the pulleys are not frozen, these windows move quite smoothly.  Unless some major incorrect repair was done on the window, it'll stop fully open, fully closed, or anywhere in between because it's in balance with the weights.   

If one pulley &amp; rope is not moving, then of course there'll be slack somewhere and the counterbalancing sash weight doesn't come into play. 

That said, they are infamous for not sealing well. That's the price of having a wooden guide with a wooden runner.   At some  point someone will have to cough up installing new energy-efficient windows and frames.