Is it permissible to write verses from the Qur’an using standard spelling for the purpose of memorization, and do these papers take the ruling of a Mus'haf, thereby necessitating ritual purity when touching them?

Firstly, the Qur'an should be written in the Uthmani script. However, there is no impediment to writing verses in a script other than this for a need, such as quoting in academic books or for educational boards for children. What is prohibited is writing an entire Mushaf in a script other than the Uthmani script. Some scholars have permitted writing in a script other than the Uthmani script when necessary. As for touching papers that contain Qur'anic verses without purity, its ruling is the same as touching parts of the Qur'an. The permissibility of doing so out of necessity has been favored, but purity is preferable to avoid disagreement.