Is it forbidden to give money to children and relatives in exchange for their care of their sick mother? And is it obligatory to pay Zakat on the mother's money without her knowledge if she refuses it?

It is not permissible to give a gift from the mother's money without her permission. If she is mentally incapacitated, it is absolutely impermissible. The children's service to their mother is obligatory upon them and does not warrant payment. As for zakat, it is obligatory on her wealth, and she must be advised to pay it if she is of sound mind, because it requires intention. If she is mentally incapacitated, zakat is obligatory on her wealth and her guardian must pay it on her behalf.