"We had lived together since the eighties."
Translation:Eravamo vissuti insieme dagli anni ottanta.
We can use either Avere or Essere, but when using Avere - Eravamo - we have to use Vissuto, and not Vissuti.
But when using Essere we have to agree with gender and singular / plural, so we have to use Vissuti.
So: Avevamo vissuto - is correct.
Eravamo vissuti - is correct.
Avevamo vissuti - incorrect.
Eravamo vissuto - incorrect.
You are correct overall, though there are some instances in which the past participle can and others in which it must agree in gender and count with "avere."
From what I recall, there must be agreement when a 3rd person direct object pronoun precedes "avere." For example, "l'ho vista" = "I saw her." Otherwise, agreement is voluntary. The below sites have more info.
https://www.italymadeeasy.com/ask018/ (Note that this one seems to have the same general idea, though it claims agreement is necessary for ALL preceding direct object pronouns, which I don't think is correct)