refactor(plex): address /code-review on the R12 S1 split

Cleanup only, no behaviour change (ruff clean + pytest 193):
- imports hoisted above `router =` and grouped/sorted (isort) across the split
  files; ruff-format pass restores the 2-blank-line PEP8 spacing the generator
  collapsed to 1.
- the shared on-disk cache root moves to a neutral app/plex/cache.py, so
  subtitles.py no longer imports it from imgcache.py (drops the cross-import).
- the former plex.py module docstring's access-model note (read endpoints open
  to any authed user; admin endpoints test/sync) is restored in the package
  __init__.py.

backend/app/routes/plex/admin.py

@@ -1,9 +1,11 @@
 """Plex admin: connection test, catalog sync, and owner watch-link config."""
 
 from datetime import datetime, timezone
+
 from fastapi import APIRouter, Depends, HTTPException
 from pydantic import BaseModel
 from sqlalchemy.orm import Session
+
 from app import sysconfig
 from app.auth import current_user
 from app.db import get_db
@@ -21,8 +23,11 @@ from app.utils import iso
 
 router = APIRouter(prefix="/api/plex", tags=["plex"])
 
+
 @router.post("/test")
-def test_connection(_: User = Depends(admin_user), db: Session = Depends(get_db)) -> dict:
+def test_connection(
+    _: User = Depends(admin_user), db: Session = Depends(get_db)
+) -> dict:
     """Admin: verify the configured Plex server URL + token, returning the server name and the
     movie/show sections (for the config library-picker). Also probes ONE real media file through
     the local mount, so a missing bind-mount / wrong path-map is caught here rather than only
@@ -41,14 +46,22 @@ def test_connection(_: User = Depends(admin_user), db: Session = Depends(get_db)
         "server_name": info.get("friendlyName") or info.get("title1") or "Plex",
         "version": info.get("version"),
         "sections": [
-            {"key": str(s.get("key")), "title": s.get("title"), "type": s.get("type"), "count": s.get("count")}
+            {
+                "key": str(s.get("key")),
+                "title": s.get("title"),
+                "type": s.get("type"),
+                "count": s.get("count"),
+            }
             for s in sections
             if s.get("type") in ("movie", "show")
         ],
         "media_check": media_check,
     }
 
-def _sample_plex_path(db: Session, plex: PlexClient, sections: list[dict]) -> str | None:
+
+def _sample_plex_path(
+    db: Session, plex: PlexClient, sections: list[dict]
+) -> str | None:
     """A real Plex-side media file path to validate the mount + path-map against. Prefers the
     already-mirrored catalog (cheap); before the first sync, pulls one movie leaf from an enabled
     section straight from Plex."""
@@ -69,6 +82,7 @@ def _sample_plex_path(db: Session, plex: PlexClient, sections: list[dict]) -> st
                 return fp
     return None
 
+
 def _media_check(db: Session, plex: PlexClient, sections: list[dict]) -> dict:
     """Resolve a sample Plex file path to its local mount path and confirm it's readable, so the
     admin sees at Test time whether the physical media is actually reachable (bind-mount +
@@ -80,12 +94,14 @@ def _media_check(db: Session, plex: PlexClient, sections: list[dict]) -> dict:
     ok = plex_paths.local_media_path(db, plex_path) is not None
     return {"checked": True, "ok": ok, "plex_path": plex_path, "local_path": local_path}
 
+
 @router.post("/sync")
 def trigger_sync(_: User = Depends(admin_user), db: Session = Depends(get_db)) -> dict:
     """Admin: mirror the enabled Plex sections into the local catalog. Synchronous (can take a
     while on a large library); a background/scheduled sync also runs periodically."""
     return plex_sync.sync(db)
 
+
 def _link_dict(link: PlexLink | None) -> dict:
     return {
         "linked": link is not None,
@@ -95,6 +111,7 @@ def _link_dict(link: PlexLink | None) -> dict:
         "last_watch_sync_at": iso(link.last_watch_sync_at) if link else None,
     }
 
+
 @router.get("/watch/link")
 def watch_link_status(
     user: User = Depends(current_user), db: Session = Depends(get_db)
@@ -103,9 +120,11 @@ def watch_link_status(
     link = db.query(PlexLink).filter_by(user_id=user.id).first()
     return _link_dict(link)
 
+
 class WatchLinkIn(BaseModel):
     enabled: bool = False
 
+
 @router.post("/watch/link")
 def watch_link_set(
     body: WatchLinkIn,
@@ -120,7 +139,9 @@ def watch_link_set(
     enabled = body.enabled
     link = db.query(PlexLink).filter_by(user_id=user.id).first()
     if link is None:
-        link = PlexLink(user_id=user.id, uses_admin=True, linked_at=datetime.now(timezone.utc))
+        link = PlexLink(
+            user_id=user.id, uses_admin=True, linked_at=datetime.now(timezone.utc)
+        )
         db.add(link)
     link.sync_enabled = enabled
     db.commit()
@@ -130,6 +151,7 @@ def watch_link_set(
         result = plex_watch.import_owner_watch_state(db, link)
     return {**_link_dict(link), "import": result or None}
 
+
 @router.post("/watch/import")
 def watch_import_now(
     user: User = Depends(admin_user), db: Session = Depends(get_db)