Refactor and optimize: deduplicate, batch queries, fix bugs

- Remove unused DATABASE_PATH import from games.py
- Extract _new_game_context(db) helper to deduplicate GET/POST error path
- Extract _parse_hole_form() helper to deduplicate hole validation in create/edit course
- Batch N+1 player handicap and tee queries in create_game into two IN() queries
- Fix hole count import bug: use max() across all tees instead of first match
- Fix migration exception handling: re-raise unless "duplicate column name"
- Cache calcSkins() result in _skinsCache; invalidate on score/skip changes
- Collapse two identical Next buttons into one in new_game.html
- Fix stale skinDoubleBackNine checkbox: reset on mode change, use x-model

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

app/database.py

@@ -139,5 +139,6 @@ async def init_db():
             try:
                 await db.execute(migration)
                 await db.commit()
-            except Exception:
-                pass  # column already exists
+            except Exception as e:
+                if "duplicate column name" not in str(e).lower():
+                    raise