[spoiler]They work as a sort of shell for the ooze, yeah, but it’s still a dead body. The body is acts more like a protective suit for the ooze than it does as a living thing. That said the ooze will repurpose/repair the bodies if they need to, this is how you end up with all the different types of zombies.
The infection can only take place prior to death, at which point it enters into a dormant state. It then aggressively mobilizes post-mortem and rapidly takes over the dead body of whatever it was living in.
As for how they go down, trauma or damage to the body will cause the ooze to go into a short dormancy state while it repairs damage. Once this state is over it will stand back up again (which is why they revive). Extreme trauma that results in a corpse unable to be repaired will result in the permanent death of that bit of ooze and the zombie staying dead.[/spoiler]
For more information I highly suggest checking out both the lab notes ingame/in the json files and checking the official lore sticky over on the library board (assuming you are fine with spoilers of course).
