What is the ruling on a father bequeathing half of his building to his mother and the other half to the needy, and what is the ruling on some heirs living in another building without paying rent, and how is the rental income distributed among the heirs?

If your mother was still your father's wife at the time of his death, then his bequest to her of half the building is considered a bequest to an heir and is not enforceable unless the other heirs consent. If they do not consent, then that half becomes part of the estate and is divided according to Islamic law. However, if she had been divorced from him and her waiting period (iddah) had ended before his death, then she is not an heir, and the bequest to her is valid, provided it does not exceed one-third of the estate.

As for his bequest of the other half to the needy in general, this is a valid bequest, and this half becomes an endowment (waqf) whose income is spent on the needy, excluding any needy individuals among his heirs. If, however, he intended the needy among his heirs, then an endowment specifically for them is not valid according to the most preferred opinion.

If the estate includes rented property, its income is divided among the heirs according to Islamic law after determining who the heirs are.