refactor: DRY improvements and type-safe error handling

DRY Improvements (api.ts):
- Extract requireApiKey() helper for API authentication
- Extract getCorsHeaders() helper for CORS header generation
- Eliminate ~20 lines of duplicated code

Type Safety (new utils/error.ts):
- Add toError() utility for safe error type conversion
- Replace all 6 `error as Error` assertions with toError()
- Proper handling of Error, string, and unknown types

Error Handling (api.ts):
- Add explicit JSON parsing error handling to all POST endpoints
- Return 400 Bad Request for malformed JSON
- Clearer error messages for API consumers

Co-Authored-By: Claude Opus 4.5 <noreply@anthropic.com>

src/utils/error.ts

@@ -0,0 +1,52 @@
+/**
+ * Error handling utilities for type-safe error conversion
+ *
+ * @module error
+ * @example
+ * ```typescript
+ * import { toError } from './utils/error';
+ *
+ * try {
+ *   // ...
+ * } catch (error) {
+ *   logger.error('Operation failed', toError(error));
+ * }
+ * ```
+ */
+
+/**
+ * unknown 타입을 Error로 안전하게 변환
+ *
+ * catch 블록에서 unknown 타입의 error를 Error 객체로 변환합니다.
+ * TypeScript strict mode에서 `error as Error` 타입 단언을 피하기 위해 사용합니다.
+ *
+ * @param error - catch 블록에서 받은 unknown 타입 에러
+ * @returns Error 객체
+ *
+ * @example
+ * ```typescript
+ * try {
+ *   await riskyOperation();
+ * } catch (error) {
+ *   // ✅ Type-safe
+ *   logger.error('Operation failed', toError(error));
+ *
+ *   // ❌ Type assertion (avoid)
+ *   // logger.error('Operation failed', error as Error);
+ * }
+ * ```
+ */
+export function toError(error: unknown): Error {
+  // Already an Error object - return as-is
+  if (error instanceof Error) {
+    return error;
+  }
+
+  // String error - wrap in Error object
+  if (typeof error === 'string') {
+    return new Error(error);
+  }
+
+  // Other types - convert to string and wrap
+  return new Error(String(error));
+}