integration: return HTTP 404 for unmatched URLs and close remaining browser-gate gaps

Co-authored-by: Cursor <cursoragent@cursor.com>

tsconfig.e2e.json

@@ -0,0 +1,8 @@
+{
+  "extends": "./tsconfig.json",
+  "compilerOptions": {
+    "outDir": "./out-tsc/e2e",
+    "types": ["node"]
+  },
+  "include": ["e2e/**/*.ts", "playwright.config.ts"]
+}