Yes, the death sentence has been put to use in Gotham.
In The Joker: Devil's Advocate, The Joker is sentenced to death, clearly indicating that in the comics, Gotham had (for at least some time) a death penalty.
After Bane successfully holds Gotham to ransom in the Christopher Nolan film The Dark Knight Rises, Dr Jonathan Crane (AKA The Scarecrow) seems to become some kind of judge and jury in a kangaroo court where Gotham's officials are tried.
When found guilty, defendants are offered the choice of exile or death. No matter which option is selected, the "guilty" party is then forced to walk across thin ice, inevitably leading to them freezing/drowning to death. As such, Dr Crane is giving the guilty parties a death sentence no matter which option they choose.
Additionally, after googling the extremely advanced phrase "batman villain armour knight helmet", the guy you ask about in the picture seems to be Prometheus.