A: While one of the most obvious ways to get rid of a Creeper attached to one of your Keepers is to destroy them both (or move them both), there ARE ways that you can destroy the Creeper while still retaining the Keeper. The key things here are 1) explicit wording, and 2) thematic appropriateness.
In Star Fluxx*, the exceptions are the Doctor, who can cure Brain Parasites, and the Engineer*, who can fix a Malfunction. In both cases they detach the Creeper, and the Keeper is left, good as new. Creeper Reassignment also specifically states that you detach the Creeper to move it. In Cthulhu Fluxx, the Dreamer states that he can detach Nightmares and discard it. Meanwhile the Sanitarium logically can cure an afflicted Keeper of Nightmares or Insanity. In Anatomy Fluxx, special actions allow you to “cure” yourself.
Some examples where the Creepers DO stay attached would be, in Star Fluxx, the weapons Laser Sword and Laser Pistol, which, again, quite logically, destroy the Keeper in order to destroy the attached Creeper. And while the Phaser in Star Trek or TNG Fluxx targets just the Creeper, it doesn’t say anything about detaching the Keeper either, so if you use it to get rid of an attached Creeper, the Keeper it’s attached to will also be destroyed. In Cthulhu Fluxx, The Feds are pretty much the equivalent, though they also destroy themselves in the process. The Necronomicon lets you move any Creeper… and says nothing about detaching it, so you’d have to move any attendant Keeper (and extra Creepers if more than one is attached).
Trash Something is the generic Action version of what the weapons allow you to do. You could trash a Keeper, and it’s Creeper will go along, or you could trash a Creeper, and it’s attached Keeper would go along. There’s no logical reason or explicit wording that lets you detach connected Creeper-Keeper combos for this Action.
*In various Star Trek Fluxxen, there are specific Engineer analogues: Scotty, Geordi, O’Brien that work the same way with respect to Malfunction.