Hello Ksenia !
1."Plus que parfait" is used to express that action A happened before action B in the same sentence.
-> Here in the sentence, it means that the Dursley have always said that they are normal, even before they started living at 4 Privet Drive.
-> Action A is "avaient toujours affirmé" and Action B is "Mr et Mrs Dursley qui habitaient au 4 Privet Drive".
2. Concerning the "Passé Conditionnel", it is usually used to express either regret or for making a supposition. In my opinion, it's used here as supposition. You could replace "imaginer" by "Jamais quiconque n'aurait supposé" : Nobody would have known or thinked that ....
3.I am not certain how to explain the use of Subjectif présent. I would say, it's here to express a wish or indignation. Please check that with a professional teacher (^_^)
3. "Dire" is used in "Conditionnel Présent" form not "Futur".
Voilà !
I hope it helps !