Are the collection of the Qur’an and ‘Umar’s addition of something new to the Adhan considered reprehensible innovations (bid’ah), or is there such a thing as a permissible "good innovation" (bid’ah hasanah) as attributed to the Prophet, peace and blessings be upon him?

Abu Bakr's compilation of the Quran is not an innovation (bid'ah), but rather a practice (sunnah) of the Rightly Guided Caliphs. The Quran was preserved in memory and on tablets, and its compilation into a single Mushaf is not to be denied. Qadi Abu Bakr al-Baqillani mentioned five justifications for Abu Bakr's action, among them: that the Prophet left it [uncompiled] for a wisdom, and Abu Bakr did it out of necessity; that it is an emulation of Allah's saying: "reciting purified scriptures, wherein are upright judgments" (Quran 98:2-3); that it is a fulfillment of His saying: "Indeed, it is We who sent down the Dhikr (the Quran), and indeed, We will be its guardian" (Quran 15:9); that its writing during his time was a reminder of its preservation and recording; and that his prohibition of traveling with it to enemy territory is evidence of its writing among the community.

As for 'Umar adding something new to the adhan, this is unknown. Perhaps what is meant is 'Uthman's addition of the first adhan for Friday prayer, which is a practice (sunnah) of the Rightly Guided Caliphs. This was justified by the people's need to be alerted to the approaching prayer due to the expansion of the city and the increased population. No legal rulings, such as prohibition or obligation, are derived from it. 'Uthman placed it in the marketplace, not in the mosque, and 'Ali did not implement it in Kufa due to a lack of need for it.