Is it permissible to use a wheelchair belonging to a government health center for Hajj for a parent who is unable to walk, given that other wheelchairs are available and it would be returned after Hajj? And is it permissible to use medical bandages and gloves belonging to the center at home for medical or non-medical purposes?

It is not permissible to take out or take home the health center's equipment without the permission of the authorized entity. Permission from an unauthorized person is not valid. The fact that your father will perform Hajj with you and is unable to walk does not permit taking the center's wheelchair; rather, you can buy or rent a wheelchair. Similarly, giving the center's gloves to a child is considered a breach of trust. None of this is permissible under the pretext of their abundance or infrequent use. Sheikh Ibn Uthaymeen, may Allah have mercy on him, was asked about using office supplies or the phone at work for personal purposes, and he replied that it is not permissible except in cases of extreme necessity, and if there is an intention to return something similar or better. He emphasized that it is incumbent upon a person who fears Allah not to use these things except with permission from those authorized to grant it, and the permission of the immediate supervisor is not valid if the general regulations prohibit it.