Please note that it’s a tricky question because I never use such an elevated language. The following are mostly based on feelings. I may delete this answer.

1. Very elevated. It’s never mandatory to use the subjonctif imparfait in any case.

2. It’s hard to think about this because ajoutai (indicatif passé simple) sounds like ajoutais (indicatif imparfait), pût (subjonctif imparfait) like put (indicatif passé simple), and the sentence itself sounds very elevated.

So I changed it to make it simpler and less ambiguous when orally said (third person subject for ajouter and plural subject for pouvoir), but keeping the tenses:

Quels mots ajouta-t-il qui pussent la pousser…

And I would better say:

Quels mots ajouta-t-il qui purent la pousser…

So I would use passé simple de l’indicatif instead of imparfait du subjonctif:

Qu’ajoutai-je qui put la pousser…

I would definitely not say pourrait (conditionnel présent) because of agreement of tenses, except if you change the sentence a bit to make the subordinate sound like it’s in the past:

Qu’ajoutai-je qui pourrait l’avoir poussée à me répondre ceci…

But you could use conditionnel passé:

Qu’ajoutai-je qui aurait pu la pousser à me répondre ceci…

You could use plus-que-parfait too:

Qu’ajoutai-je qui avait pu la pousser à me répondre ceci…

Or even futur antérieur:

Qu’ajoutai-je qui aura pu la pousser à me répondre ceci…

And why not passé antérieur:

Qu’ajoutai-je qui eut pu la pousser à me répondre ceci…

3. I’m not sure to understand the English sentence perfectly. What I know is that (B) is correct and means you are not sure you really saw them the day before yesterday.

(A) would sound better with vis-je than ai-je vu and I don’t feel like it carries the idea of a doubt. It still may be correct with ai-je vu.

4. Pourrait is conditionnel présent when your link is about conditionnel passé.

First form of conditionnel passé is:

(il) aurait pu

I already mentioned it in point 2.

Second form is:

(il) eût pu

I would say it’s correct to use it too.