Is it permissible for the questioner to remain in a house owned by the state after his father's retirement from work and his older brother's taking over the job on his behalf, knowing that the brother resides outside the house and the questioner's financial situation is good?

We do not know the conditions set by the state for the handover of its houses to employees, and Muslims are bound by their conditions. If the state permits the employee to allow whomever they wish to reside in the house, then it is permissible. However, if it stipulates that only specific individuals may reside in it (such as children, parents, and spouse), then these conditions must be adhered to, and you must vacate the house if you are not among those permitted to reside in it.