Does Islam care about antiquities and museums in all their forms, or are there jurisprudential controls for that?

The Prophet, peace and blessings be upon him, used to seclude himself for worship in the Cave of Hira before his prophethood, and the revelation descended upon him there. Then he left it. There is no narration that the Prophet, peace and blessings be upon him, or his companions, visited it for worship or to seek blessings from it, as they were the keenest of people to do good. If there had been any good in it, they would have preceded us to it.

It is permissible for a Muslim to visit the Cave of Hira for the purpose of remembrance and reflection, as long as there is no fear that others might imitate them in a negative way. However, visiting it for worship, undertaking a journey to it, or believing it to be a Sunnah, is an innovation that contradicts the guidance of the Prophet, peace and blessings be upon him, and his companions.

As for museums that contain statues, there is no good in them or in visiting them, due to the presence of carved images and religious transgressions.