Is there anyone who can act on behalf of the ruler in carrying out the حد (prescribed punishment) for sodomy in lands where Allah’s law is not applied, and is burning or throwing from high mountains considered a حد and an expiation, or a حد in this world followed by Allah's punishment even with repentance?

The implementation of the legal punishments (Hudud) is the prerogative of the Muslim ruler or his deputy, due to their importance and seriousness, and to avoid the chaos that might ensue if every individual were permitted to implement them. If Muslim rulers do not implement the Hudud, then Muslims must advise them and call upon them to apply Allah's law. And if a legal punishment is carried out on a person, it serves as an expiation for him.