Have the authorities of the Russia-recognized republics of Donetsk and Luhansk commented on whether they consider themselves bound by the Geneva Conventions? They didn't sign any of those.
The fake "republics" cannot be party to any international convention, such as the Geneva Convention. You have to be a country to be able to do that. The so-called "republics" cannot do that for the same reason as you and I cannot use Russian help, secede from the United States, and declare that we are bound by the Geneva Convention (and then violate that convention at every step).
Do they (LPR/DPR) maybe consider themselves successor states of the USSR, implicitly bound that way?
The fake "republics" have issued no statements as to whether they consider themselves successor states of the USSR. In fact, their poorly defined legal status has been exploited by Russians. In Russia, there is still a moratorium on death penalty. LPR and DPR have no such moratorium. This allows Russia to hand over the captured soldiers and sentence them to death by the kangaroo courts of LPR and DPR. Geneva convention means next to nothing to the invading Russian army. It is expected that Geneva convention will mean even less to the fake republics.
REFERENCES:
UNITED NATIONS, June 9. /TASS/. The United Nations calls for compliance with the Geneva Conventions on the treatment of prisoners of war, Spokesman for the UN Secretary General Stephane Dujarric said at a briefing on Thursday, commenting on the death sentence handed to mercenaries from the United Kingdom and Morocco in the Donetsk People’s Republic (DPR).
"First of all, we stand against the death penalty, we always have and always will. We call on all the combatants, who have been detained, to be afforded international protection and to be treated according to the Geneva Conventions," he pointed out.
A Donetsk court on Thursday found British nationals Shaun Pinner and Aiden Aslin and Moroccan Brahim Saadoun guilty of participating in military activities as mercenaries on the side of the Ukrainian Armed Forces. According to the DPR Prosecutor General's Office, the testimony of Shaun Pinner, Aiden Aslin and Brahim Saadoun confirmed their involvement in crimes as defined by Article 34.2 of the DPR Criminal Code (crimes committed by a group of people), Article 323 of the DPR Criminal Code (forcible seizure of power or forcible retention of power) and Article 430 of the DPR Criminal Code (mercenary activities). The decision can be appealed within a month.
UN calls for compliance with Geneva Conventions following DPR court’s ruling: https://tass.com/world/1463091
Responding to the news that a separatist “court” in Russia-occupied Donetsk ordered the death of two captured two British nationals and a captured Moroccan national, Denis Krivosheev, Amnesty International’s Deputy Director for Eastern Europe and Central Asia, said:
“This is a blatant violation of international humanitarian law on so many counts.
“The three were members of the Ukrainian regular forces and under the Geneva Conventions, as prisoners of war, they are protected from prosecution for taking part in hostilities. The only exception is prosecution for alleged war crimes, in which case there must be sufficient admissible evidence, and fair trial standards must be ensured. Not only this is not the case in this scenario – they were not tried by an independent, impartial regularly constituted court but by Russian proxies. The so-called ‘charges’ against them would not constitute war crimes. And most outrageously of all, the taking of their lives as result of the grossly unfair proceedings would constitute arbitrary deprivation of life.
“In fact, wilfully depriving a prisoner of war or other protected person of the right to a fair and regular trial constitutes a war crime. Russia, as the occupying power, bears responsibility for the treatment of all prisoners of war and others deprived of their liberty. They must ensure this so-called ‘sentence’ is immediately quashed, and that these men are treated in full compliance with international humanitarian law.”
Ukraine/Russia: “Death sentences” against three foreign members of Ukrainian forces by separatists’ “courts” a blatant violation of international law: https://www.amnesty.org/en/latest/news/2022/06/ukraine-russia-death-sentences-against-three-foreign-members-of-ukrainian-forces-by-separatists-courts-a-blatant-violation-of-international-law/
Note: The relevance to the real world of whether any non-country entity considers itself bound by the Geneva Conventions is left as an exercise to the reader. What is important for the real world is (a) whether the entity is in fact a sovereign country (and not a fictional entity), and (b) whether it does in fact observe the Geneva Conventions (and not just declares that it does, and then violates them).