Council of Elrond

Does a tapped haven (or darkhaven/wizardhaven) site untap during the untap phase?
e.g. if I played a card that taps the site at a haven, do I need to move away (returning the haven to the location deck rather than the discard pile) and then move back again to play another card that taps the site? (or alternatively use some site untapping effect).

Alternatively, since I can have multiple copies of havens in my location deck, if I have a company at a tapped Lorien, could I play another copy of Lorien when I declare movement for them to "move" to?

Havens and other sites do not untap during the Untap Phase because the rules state "untap all of your other tapped non-site cards" (METW p. 68).

There are ways to get an untapped Haven because you may have multiple copies in your location deck. Ways to untap havens are discussed here: viewtopic.php?f=16&t=4172&hilit=haven+untapping

You can get an untapped Haven card by:

• Playing a new character under General Influence at the Haven either (A) before your Wizard is in play or (B) when your wizard is in play and is at the same Haven (i.e., in a different company). This new character cannot join any existing company at the tapped haven on the turn he was played because otherwise that character should have been played with that company in the first place. This is because of the rule that "Company composition changes, including bringing a character into play, must all be done at the same time during the organization phase."
• Moving a company to the Haven (and after all M/H phases, choosing to not join with an existing company at that Haven).
• Splitting an existing company at a Haven. Both companies can stay. Or, one of the companies can move away from the tapped Haven and another company can stay at the untapped Haven.