Is it proven that Abu Bakr asked the Prophet (blessings and peace of Allah be upon him) “Has the Shaytaan died?”?
We do not know of any hadeeth in these words or with a similar meaning. These are words that must have been fabricated and falsely attributed to the Messenger of Allah (blessings and peace of Allah be upon him), for which there is no basis, because the Prophet (blessings and peace of Allah be upon him) knew that Allah, may He be exalted, has given the Shaytaan respite until the Day of the time appointed. Allah, may He be glorified and exalted, says (interpretation of the meaning):
“(Iblees) said: ‘Allow me respite till the Day they are raised up (i.e. the Day of Resurrection).’
(Allah) said: ‘You are of those allowed respite.’”
[al-A ‘raaf 7:14, 15]
“(Iblees (Satan)) said: ‘O my Lord! Give me then respite till the Day they (the dead) will be resurrected.’
Allah said: ‘Then, verily, you are of those reprieved,
‘Till the Day of the time appointed.’”
What is meant by the Day of the time appointed is the day when all creatures will die at the first Trumpet blast, according to the most correct scholarly opinion.
For more information on that, please see the answer to question no. 10034
Allah has extended the life of Iblees until the time of the first Trumpet blast, so he will remain alive until that time and will not die before it. That is so that what Allah, may He be exalted, has decreed by His wisdom and knowledge may be completed, namely to test and try all people so that He might requite them on the Day of Resurrection for their deeds.
‘Abdullah ibn Ahmad narrated in Zawaa’id az-Zuhd (p. 266) that Salaam ibn Miskeen said: al-Hasan – i.e., al-Basri – was asked: O Abu Sa‘eed, does Iblees sleep? He said: If he slept, we would have a rest.
The hadeeth mentioned in the question is a false hadeeth for which there is no basis.
Please see the answers to questions no. 34725 and 135763 for more information on the seriousness of telling lies about the Messenger of Allah (blessings and peace of Allah be upon him) and the prohibition on narrating false things from him, and the ruling on that in sharee‘ah.
And Allah knows best.