Why is cyclopropane-1,1-diol stable
The bonds in cyclopropane 1,1-diol are less strained than the cyclopropanone. 

The preferred/lowest energy for tetrahedral bond angles is 109.5 degrees so the closer they can get to that the lower the energy. In cyclopropanes the angles are 60.
The lowest energy angle for the R2C=O system (SP2) is 120 degrees, in cyclopropanone it is 60. This is a very strained and unstable system, anything that reduces that strain is a positive change energywise.

This problem is discussed here see problem 2