Qur'anic Verse

"Imam mubin" ("manifest imam", "clear imam", or "clear record") is generally understood to refer to a divine record in which all things and divine decrees are inscribed.

(Some exegetes have explained that it is called the "imam" of divine records because it is the highest of the books, hence the leader of books, as opposed to lesser divine records, such as individuals' scrolls of deeds.)

Some hadith also say that the "imam mubin" is Imam Ali. In one hadith, Imam Ali is quoted as saying that, "I am the imam mubin. I distinguish between truth and falsehood, and I have inherited this from the Messenger of Allah."

Perhaps both views are correct, in that it is possible for Allah to provide any of His servants with knowledge of all things.

Some hadith offer explanations for the circumstances of revelation for this verse. However, they do not relate to the phrase "imam mubin" but rather refer to why the previous phrase ("what they have sent ahead and their effects [which they left behind]") might have been revealed. These relate to the community in Medina as well as the general idea that people are rewarded or punished after death for the good or bad practices or legacies they have left behind. However, they do not say anything specific linking the word "imam" directly to the occasion of revelation. I am not aware of any hadith that do this, although there may be some!
 

Historical sources list 4 daughters for the Prophet (S). It is not clear whether all of them were his biological daughters, or only one (namely, Fatimah al-Zahra).

If he only had one biological daughter, the other 3 girls who are mentioned in some sources as his daughters would have been girls that he was raising (as adopted daughters), and it would be reasonable to refer to them all generally in this context as "daughters". 

Using a plural also does not preclude daughters in the future, whether they be biological daughters (which didn't happen) or step-daughters (which would have been an open possibility given that he remarried after Hazrat Khadijah). However, if the verse had only said "daughter", and he only had one daughter, it would have been a specific instruction for a specific person and not a general instruction.

Also, the Qur'an occasionally uses a plural form to indicate generality, not multiplicity. For instance, the verse of mubahilah instructs the Prophet to take "our selves" and "our women" (in the plural) to the meeting for mubahilah, but he only brought one person as his "self" (Imam 'Ali) and one person as "his women" (Fatimah al-Zahra'). 

Lastly, a prophet can be considered a father of his people (as in "I and 'Ali are the fathers of this ummah"), and so referring, in general, to the girls of the community as his "daughters" would not be unreasonable, particularly since the verse also addresses the "believing women", although admittedly this is not the interpretation that first comes to mind. 

Historical matters can be complicated. It can be difficult to know the exact details of what happened over a thousand years ago. One has to have faith that Allah has preserved what is necessary for us to know, and in this case the emphasis is on the spirit of the ayah rather than the specifics of lineage.