Learn English – What’s missing in “The discovery of magnetic effects of coals made possible to measure an electric current”

Here prior is being used as a shorthand of sorts for a fuller phrase such as "prior to meeting him here" or "prior to coming into the room". It is technically not correct usage, but it is probably relatively common in some areas. (Since I don't know where "Bazan", the speaker, is from, other than 'not from my area', I can't really say how common it might be; but it's not common in my area, at least.)

"Going about his day", as noted in the comments, is an idiom meaning to do all the normal, run-of-the-mill, routine things that you usually do every day; shower & shave, eat breakfast, read the papers, check your e-mail, get ready for work, and so forth.

Of the two choices, the simple past is best (although not "official!")

She said she would collect the book on Monday when she came home from school.

Irrealis conditionals like would call for this kind of tense simplification in subordinate clauses. We use the preterite came here because at the time in the future when the subject will collect the book, the action of coming home is already completed.

In Swan's Practical English Usage, the author says:

If the main verb of a sentence makes it clear what kind of time the speaker is talking about, it is not always necessary for the same time to be indicated again in subordinate clauses.

And also:

Would, like will, is avoided in subordinate clauses; instead, we generally use past verbs. This happens in if-clauses, and also after most other conjunctions.

Contrast your example with the same sentence using the simple future instead of the conditional:

She said she will collect the book on Monday when she comes home from school.

Here the verb in the WH-clause is simplified to the simple present.