Format the panel and admin date and clock in the stored application timezone

The panel snapshot now carries the admin-saved `timezone` ("" when unset);
the coordinator threads it onto the household due-today sensor, and the
Party Board and Quest Log cards format their kicker, header and dock
date and clock in it, falling back to Home Assistant's zone only when
none is stored. The admin app loads the stored zone once
(AppTimezoneProvider) and uses it for completion clocks, History day
grouping, clocks and lateness, and "today" in Schedule, History and new
task defaults; Settings publishes a saved zone app-wide at once.

Resolves review finding TZ-001 (task 7e173881).

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

tests/test_api_admin_snapshot.py

@@ -165,7 +165,9 @@ async def test_admin_jwt_returns_documented_snapshot(
 
     Every field is asserted as an explicit literal, with ONE clock
     read for the request.  With no missed view in the real snapshot,
-    the admin body equals the panel body for the same household.
+    the admin body equals the panel body for the same household — the
+    panel body additionally carries the stored ``timezone`` its cards
+    format the visible date and clock in (unset here).
     """
     client = snapshot_household.client
     seed = snapshot_household.seed
@@ -241,7 +243,7 @@ async def test_admin_jwt_returns_documented_snapshot(
         "/api/v1/panel/snapshot", headers=_panel_headers()
     )
     assert panel.status_code == 200
-    assert panel.json() == body
+    assert panel.json() == {**body, "timezone": ""}
 
 
 async def test_admin_keeps_missed_instance_panel_omits_it(