Hello, really sorry for the late response.
Either vol. 4 of The Cambridge History of Christianity: Christianity in Western Europe c. 1100-c. 1500, ed. Miri Rubin & Walter Simons, Cambridge: Cambridge UP, 2010 or The Oxford Handbook of Medieval Christianity, ed. John H. Arnold, Oxford: OUP, 2014 is the excellent collection of essays with high academic standard, though these two might be a bit expensive and not primarily intended for beginners.
- If you prefer one-volume introductory overview work of (high) medieval Christianity written by a single author, I'd still recommend Richard W. Southern's Western Society and the Church in the Middle Ages, rev. ed. Harmondsworth: Penguin, 1990.
- Robert I. Moore's The Formation of a Persecuting Society (1987) is still the classic of the transformation of Western society as well as the Latin Christendom in the 12th and the 13th century, and listed also among the AH's book list. His War on Heresy (2012) might also be not so bad choice, since Christine Caldwell Ames' Medieval Heresies might be a bit difficult. Alternatively, Jennifer K. Deane, A History of Medieval Heresy and Inquisition, Lanham: Rowman & Littlefield, 2011, is also a readable introduction.
- As for the church reform movement, Kathleen G. Cushing, Reform and the Papacy in the Eleventh Century: Spirituality and Social Change, Manchester: Manchester UP, 2005, is probably the easiest read in English, but if you can read German, I'd instead recommend: Claudia Zey, Der Investiturstreit, München: C. H. Beck, 2017.
- Concerning Popes and the Papacy, I still vastly prefer Ian S. Robinson, The Papacy, 1073-1198: Continuity and Innovation, Cambridge: Cambridge UP, 1990, to other books like Collin Morris, The Papal Monarchy (1989), though Robinson's book does not primarily cover the pontificate of Innocent III (r. 1198-1216).
I'd also be willing to add any relevant topic, so if you feel that the book on any relevant topic still lacking in the list, please don't hesitate to make a notice to me.