Why do unbonded electrons exist in pairs?
A single unpaired electron is called a radical, and usually aren't stable. Each electron orbital can hold two electrons, one spin up, other spin down. So a lone pair is just a full electron orbital, and cannot accept another electron. An orbital with 1 electron will readily react to get another electron and form a pair. 

Check this: Free Radical

The first reaction they show is UV light being used to break $\mathrm{Cl}_2$ gas into $2\mathrm{Cl}$ radicals, each with a single lone electron, forming free radicals. These radicals could be used as free radical polymerization initiators, or for other reactions.

In the structure you show, the molecule forms a trigonal planar configuration, with the lone pair forming the third arm. If you could stably achieve two free radicals instead of the lone pair, it would have tetrahedral configuration, with the radicals forming 2 of the 4 arms.

The theory behind it is more quantum mechanics than I can remember, but I know it involves the Pauli exclusion principle and Hund's rule.