Is it permissible for a public official delegated to the board of directors of a foreign company that has an agreement with his original institution to enjoy in-kind benefits (such as a mobile phone, car, or health insurance) provided to him by the foreign company, and is it permissible for him to use the work phone for his personal calls?

It is not permissible to use work tools for personal purposes except with the permission of the employer or whoever has the authority to grant permission. If the company allows unrestricted use of the mobile phone and car, then you may do so; otherwise, their use must be restricted to the benefit of the work only. There is a difference between private companies, where the use of work tools for personal purposes is permissible with the employer's permission, and the public sector (government), where it is not permissible, even if authorized by an official, because it is public property and no one has the authority to permit its personal use.