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/src/auth/auth.controller.ts

@@ -1,4 +1,13 @@
-import { Body, Controller, Get, Headers, Post, Query, Req, Res } from "@nestjs/common";
+import {
+	Body,
+	Controller,
+	Get,
+	Headers,
+	Post,
+	Query,
+	Req,
+	Res,
+} from "@nestjs/common";
 import { ConfigService } from "@nestjs/config";
 import { Throttle } from "@nestjs/throttler";
 import type { Request, Response } from "express";