Is it permissible not to give needy siblings their money unless absolutely necessary, and is it permissible to give out the zakat on their money without their knowledge to those among them who are eligible to receive it?

The decisive factor is the mutual consent between the agent and the principal. The agent is not permitted to act except within what he has been explicitly or customarily authorized to do.

The agent can try to convince the principal to leave his dues with him for investment. However, if the principal refuses, the agent is not allowed to withhold his money or place an embargo on it.

Regarding the issue of Zakat, the principal, whose wealth has reached the nisab (minimum threshold) and a full year has passed over it, must authorize the agent to pay Zakat on his behalf. If some of them are eligible for Zakat due to their poverty, there is no harm in giving it to them, unless the principal has specified a particular person to whom it should be given, in which case it is not permissible to contravene his instruction.