Is not specifying an age for the number of Tarawih rak'ahs as 20, even though the Prophet did not specify a number, considered a bid'ah (innovation)? And why was that permissible for 'Umar, while it is not permissible for others?

Innovation (Bid'ah) is everything that is newly introduced into the religion for which there is no evidence, or which the Prophet (peace be upon him) left, despite its cause being present and no impediment preventing it. There is no such thing as a "good innovation" (bid'ah hasanah) in a legal sense; rather, every innovation is misguidance. What some of the early Muslims (Salaf) meant by "good innovation" was a linguistic innovation, meaning something new.

The construction of minarets falls under the category of Maslaha Mursala (public interest not explicitly addressed by texts but consistent with the general objectives of Islamic law). It has a basis in the call to prayer (Adhan) of Bilal (may Allah be pleased with him) from the rooftops of some houses adjacent to the Prophet's Mosque. It is recommended for the purpose of conveying the muezzin's voice and identifying the mosque.

Regarding Tarawih prayer and night prayer in general, the Prophet (peace be upon him) did not specify a number of rak'ahs, but rather said: "Two by two." So, if a person prays twenty, thirty, or a hundred rak'ahs, there is no harm. The difference in the number of rak'ahs between the Prophet's actions and the actions of the Rightly Guided Caliphs is not considered an innovation, because the prophetic sayings (Sunnah Qawliyyah) take precedence over his actions (Sunnah Fi'liyyah) when there is a conflict, although here there is no conflict. This is supported by the Prophet's saying: "Hold fast to my Sunnah and the Sunnah of the Rightly Guided Caliphs."