chore: update lint scripts and improve formatting consistency

- Added `lint:fix` scripts for backend, frontend, and documentation.
- Enabled `biome check --write` for unsafe fixes in backend scripts.
- Fixed imports, formatting, and logging for improved code clarity.
- Adjusted service unit tests for better readability and maintainability.

frontend/package.json

@@ -7,6 +7,7 @@
 		"build": "next build",
 		"start": "next start",
 		"lint": "biome check",
+		"lint:write": "biome check --write",
 		"format": "biome format --write"
 	},
 	"dependencies": {