Why is cis-1,2-dichloroethene more stable than trans-1,2-dichloroethene?
Here's a pre-MOT rationalisation of the cis form being more stable than the trans form for 1,2-dihaloethylenes:

The lone pair of chlorine atoms is involved in resonance with the double bond, as it does so positive charge appears on one chlorine and negative on another. A cis geometry allows for a stabilizing interaction (attraction) between these positive and negative charges, which makes this resonance form a greater contributor to the resonance hybrid than it is for the trans isomer.



MOT-based explanation: 

Eyring's work suggests that there is maximum delocalisation possible in trans form.

Quoting, from the same paper: 


  In a recent paper Binghams proposes that the main effect responsible for the “cis effect” of ethylenes is an effect which tends to destabilize the trans form rather than provide extra stabilization for the cis form. In short, this is an unfavorable conjugation effect which stems from the fact that the conjugation in these cases is between two filled orbitals (halogen lone pair and $\pi$ bond). The interaction gives rise to two new orbitals, one bonding and one antibonding, the latter being more antibonding than the former is bonding. Consequently, the total interaction is energetically unfavorable, in contrast to interaction in hydrocarbon polyenes. This would be equally true for both the cis and trans forms but, following Eyring’s argument, the delocalization is greater in the trans form.
  Consequently this form will be the less stable one. 




Reference: Structure of 1,2-difluorocyclopropane and the "cis effect"