Is it permissible to collect our personal Zakat on money to purchase a property for a sick uncle suffering from schizophrenia, so that he may rely on it for his livelihood, and is Zakat obligatory on this collected money if it remains with us for more than a year?

It is permissible to give a poor person from zakat funds what he can use to purchase a property for his sufficiency. This is what the Shafi'is permitted for non-professionals. The money can be given to the poor person to buy it himself, or the zakat giver can buy it for him with his permission, or the ruler or his guardian can buy it. As for zakat for relatives, if the poor brother is obliged to be supported by his wealthy brother, it is not permissible to give him zakat; rather, support must be provided to him. However, if his support is not obligatory, then it is permissible to give him zakat. Zakat must be disbursed and given to the poor person immediately; it is not permissible to withhold it.