Is it stipulated that the entire Qur’an be recited in Ramadan for it to be counted as a completion in that month, or is it permissible to read parts of it before Ramadan and complete it therein?

There is no evidence commanding to start reading from the beginning of the Mus'haf upon the advent of Ramadan. Rather, it is recommended to continue reading until one completes it, and then begin another completion. It is recommended for the Muslim to continue his current completion; if he completes it before Ramadan, he should start another one, and if he has not completed it, he should finish it and then start anew. Completing the Qur'an in Ramadan is preferable, but if one is unable to do so, there is no sin, as it is not obligatory.