feat(printers): airduct mode + status badges + force refresh on printer card

Surface four Home Assistant-style controls on the Printers page card:

  - SD Card badge in the top status row (green / red, icon-only).
  - Enclosure Door badge in the top status row (green / yellow, icon-only).
    Detection per printer family — X1/X1C/X1E read home_flag bit 23, all
    others read top-level `stat` (hex string) bit 23 — so X1 firmware that
    does not flip stat bit 23 stops false-triggering "open". WebSocket
    status-change dedup key now includes door_open so toggling the door
    alone publishes a push, no 30s REST-poll wait.
  - Airduct Mode badge beside the speed control (cooling / heating)
    for P2S/H2D/H2C/H2S; one-click dropdown calls the existing
    set_airduct MQTT command via a new POST /printers/{id}/airduct-mode
    route.
  - Force Refresh entry in the kebab menu — calls the existing
    /printers/{id}/refresh-status endpoint to request a pushall snapshot
    without forcing a reconnect.

  Tests: door-open parsing (X1 home_flag, non-X1 stat, ignore mismatched
  source, invalid hex) and airduct route (validation, not-connected,
  success, failure).

backend/app/api/routes/printers.py

@@ -583,6 +583,7 @@ async def get_printer_status(
         ipcam=state.ipcam,
         wifi_signal=state.wifi_signal,
         wired_network=state.wired_network,
+        door_open=state.door_open,
         nozzles=nozzles,
         nozzle_rack=nozzle_rack,
         print_options=print_options,
@@ -2333,6 +2334,33 @@ async def set_print_speed(
     return {"success": True, "message": f"Print speed set to {speed_names.get(mode, 'Unknown')}"}
 
 
+@router.post("/{printer_id}/airduct-mode")
+async def set_airduct_mode(
+    printer_id: int,
+    mode: str = Query(..., description="Airduct mode: 'cooling' or 'heating'"),
+    _=RequirePermissionIfAuthEnabled(Permission.PRINTERS_CONTROL),
+    db: AsyncSession = Depends(get_db),
+):
+    """Set the airduct mode (cooling/heating) on supported printers (P2S/H2*)."""
+    if mode not in ("cooling", "heating"):
+        raise HTTPException(400, "Mode must be 'cooling' or 'heating'")
+
+    result = await db.execute(select(Printer).where(Printer.id == printer_id))
+    printer = result.scalar_one_or_none()
+    if not printer:
+        raise HTTPException(404, "Printer not found")
+
+    client = printer_manager.get_client(printer_id)
+    if not client:
+        raise HTTPException(400, "Printer not connected")
+
+    success = client.set_airduct_mode(mode)
+    if not success:
+        raise HTTPException(500, "Failed to set airduct mode")
+
+    return {"success": True, "message": f"Airduct mode set to {mode}"}
+
+
 @router.post("/{printer_id}/chamber-light")
 async def set_chamber_light(
     printer_id: int,