When I have reanimated bodies and decide to fast travel or go through doors triggering the load screen, my minion is disintegrated. I believe that's because the game actually removes the actor and re-creates it in the new loaded area, and this removes all effects currently on the minion (such as the secondary effect with disintegration script). Also, any buff I have on the minion is gone, for the same reason.

Question is: What can I do to keep my reanimated bodies as they were before loading the new area? I don't think I can avoid the removal/re-creation, but what can I do about the effects in place? Any workaround at least to avoid bodies being disintegrated?

In short, the Reanimate archetype is completely fubar. Your theories are correct. All spells and abilities directly applied to a reanimated actor are dispelled/stripped from them when the player goes through a load screen (in my testing, not all load screens cause this but many do).

The disintegration is easy to fix. You need to remove the secondary reanimate effect from all the reanimate spells. You'll also find that the bonus from the Dark Souls perk gets stripped from them as well...

I have found a workaround to the "Dispel Everything" that seems to happen during load screens but is very involved and not for the faint of heart. You have to create a quest with a number of aliases and assign each new reanimated minion to an alias. You can then interact with the Reference Alias which, in my testing, is immune to the "Dispel Everything" bug.