Is the eldest brother's management of his deceased father's estate on behalf of his siblings, without dividing it for years, permissible? And how is the Zakat for this estate to be paid for the past years, and is it paid as if it were a single owner's wealth, or does each heir pay Zakat on their share if it reaches the Nisab?

If the children are adults and of sound mind, they may authorize their brother to manage their money with their consent. If one of them requests his share, it must be given to him.

However, if the children are minors, the person responsible for their money is the father's guardian, or the ruler, or whoever the ruler appoints. If none of these exist, then it is the trustworthy administrator, provided that his management is in the best interest of the minor.

Regarding Zakat: Zakat is not obligatory on real estate intended for personal use, nor on trade goods, unless the owner intended it for trade at the time of acquisition. As for cash, Zakat is obligatory on it at the end of every lunar year if it reaches the threshold (nisab), and each heir must pay the Zakat for his share.