Is an "ijazah" (certificate) in the mastery of recitation (tilawah) and tajwid, which did not include reading the entire Qur'an to the issuer (mujiz), considered a legitimate "ijazah"? And is it permissible for the one who received this "ijazah" to grant "ijazah" to others if they recite the entire Qur'an to him?

What you are asking about is termed by scholars as ijāzah (certification) in the Quran and recitation of parts of it. This is a type of ijāzah that requires mastery and precision. If you are proficient and qualified, your ijāzah is valid for what you have recited to the shaykh only. It is not permissible for the shaykh to issue you an ijāzah for reciting the entire Quran if you have not recited it in full, because that is deception, and scholars have warned against it. If you recited to the shaykh from the mushaf (physical copy) and not from memory, then it is not permissible for you to grant ijāzah to others, as this is a specific ijāzah not granted to others. Furthermore, listening to an audio recording is not considered valid for obtaining an ijāzah, because not everyone who listens can perform like one who has been granted ijāzah.