What are the supplications or verses that help in future marriage, what are the criteria for choosing a suitable partner, and can the Quran be a guide in answering this?

There is no specific supplication (du'a) for marriage fixed from the Prophet, peace and blessings be upon him. However, it is recommended to supplicate that Allah bless you with a righteous spouse, and to perform the Istikhara prayer when approaching marriage.

As for seeking guidance by opening the Qur'an randomly, it is forbidden and among the ways of devils. While seeking guidance from it by following its rulings is the path to guidance and happiness.