What is the origin of Ahmad's doctrine regarding the validity of repentance while persisting in another sin, and does that mean the invalidity of marriage after repentance from adultery? And if he has two opinions on the matter, which of them is more preponderant?

Repentance from a sin is valid even with persistence in other sins. This is the view of the majority of scholars, and Imam An-Nawawi related a consensus on this, although Ibn Al-Qayyim mentioned a difference of opinion on the matter. What was reported from Imam Ahmad in this context is interpreted as a deterrent and a means of blocking pretexts, not as an agreement with the Mu'tazilah. The imams of his school understand his words to mean that it is not a general repentance, not that the sin of the repentant is like the sin of one who persists in major sins. The repentance of one who repents from zina (fornication/adultery) is valid even if he persists in other sins, and his marriage is valid if it fulfills its conditions.