Is it permissible to act upon the father's will, which stipulates allocating half of a building to his mother and the other half to eligible beneficiaries from the rent, and housing some of the heirs in the second building without paying rent? And is it permissible to delay selling the buildings and distributing the inheritance throughout all these years?

It is permissible to bequeath the rent of half the building to any needy heir, provided it does not exceed one third of the estate. This is a bequest to وقف (endow) this half and specify its use. As for bequeathing half the building to your mother, this is a bequest to an heir and is not enforceable unless the other heirs consent. If the heirs agree to some of them residing in the building, whether for rent or for free, there is no harm. However, if the heirs who do not reside there demand their share of the rent, it is their right. If one of them desires his share and does not agree to the building remaining in this state, then his share of the inheritance must be given to him, either by selling the property or by some of the heirs purchasing his share.