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.

backend/package.json

@@ -13,7 +13,7 @@
 	"scripts": {
 		"build": "nest build",
 		"lint": "biome check",
-		"lint:write": "biome check --write",
+		"lint:write": "biome check --write --unsafe",
 		"format": "biome format --write",
 		"start": "nest start",
 		"start:dev": "nest start --watch",