What is the ruling of Islamic law regarding how to deal with the estate of a deceased brother, specifically the house and car, and what are the implications for the speaker's use of these assets, such as paying past rent or purchasing them outright, and how are they to be valued, taking into account any improvements made by the speaker?

You must pay your brother's heirs rent for your use of the house and car since his death, because the inheritance has become theirs. You are not entitled to use these things even if you are a guardian of orphans or in need. The mother has no authority over the children's money unless she is appointed as a guardian. What you built in your brother's house, you are entitled to its value as dismantled if you built it for your own need, and its value as standing if you built it for the benefit of the children and with the permission of the heirs. The rent is to be assessed by experts, and in case of dispute, the matter is referred to the Sharia (definition) court.