Stall torque for servo
But does it mean it'll actually drop the load at any distance greater
  than 1cm? That is, it'll drop the load at distances 1.1cm, 1.2cm, 2cm,
  etc.?


Generally yes, assuming the stall (holding) torque specification is accurate. 

A servo with worm gearing may be able to hold its position even with no power. However even a worm drive can slip if enough force is applied. If the gearing supports a certain holding torque unpowered then it should support even more holding torque when powered (assuming it doesn't jump teeth or break).

Another definition of stall torque is the load that causes it to stall. In this case at higher torque the servo may continue holding its position, or it may be pulled off until the positional error drives the motor harder (typical behavior of 'analog' hobby servos), or it may 'drop' the load until torque reduces, or it may shut down and move freely due to detecting an overload (behavior of some 'digital' hobby servos).

Note that the torque specifications of 'hobby' servos are often 'optimistic', and may even burn the servo out if an attempt is made to operate it at rated torque.