Adopt Home Assistant's time zone for the API household clock

The deployed API host runs UTC and the household `timezone` setting was
never set, so the panel snapshot read UTC: morning quests showed overdue
hours early and `today_iso` rolled over at UTC midnight.

The integration now reports `hass.config.time_zone` on every panel
request (`X-NestQuest-Timezone`).  While the admin has not chosen a zone
(`timezone_configured` is false) the API adopts and persists it, so the
panel, the admin plane, and the missed-sweep scheduler all read the
household clock out of the box.  An explicit admin choice — including an
explicitly chosen empty zone — still wins; an invalid reported zone is
logged and ignored.  No schema change.

Task: 6351b36d-0eef-4e7d-bf5a-96b754cbc883

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

tests/test_coordinator.py

@@ -686,6 +686,8 @@ async def test_coordinator_configured_token_builds_real_client(
         hass, entry, session_factory=_session_factory
     )
     assert isinstance(real_client, NestQuestApiClient)
+    # Home Assistant's zone is reported to the API (task 6351b36d).
+    assert real_client._household_timezone == hass.config.time_zone
 
     set_coordinator_client(entry, real_client)
     assert await async_setup_entry(hass, entry) is True