Council of Elrond

... or: why not?

CRF, Rulings by Term, Active Cinditions wrote:Your opponent's resources may be the active conditions for your resources, but may
not be the targets for your resources.

CRF, Rulings by Term, Targets wrote:You cannot target an opponent's character or resources with your own resources.

What stops me from playing the Hidden Haven on opponent's (e.g. to prevent an opponent's company from playing anything playable as written on the site card)?
What stops me from playing the Houses of Healing on opponent's ?

I hope that there is a rule that stops me from doing so, but I cannot find it.

What about an opponent's company (rather than characters)?

(Almost) the same. (Almost) nobody will complain about Fellowship, but some will be unhappy with Palm to Palm (some alternative to So You've Come Back).

Here's fun one: Play People Diminished on an opponent's Free-hold and force him to face those attacks without being able to play resources to help him.

It is not sure which company is "the company [that] faces attacks".