Update lit-starter-ts to lit-next

* Upgrades the source to work with lit-next
* Aligns the linting and formatting rules/settings & ignores with the monorepo
* Upgrades to the same version of typescript and tsconfig settings as the monorepo
* Upgrades the test runner from karma to web-test-runner
* Upgrades the dev server from es-dev-server to @web/dev-server

.eslintrc.json

@@ -1,4 +1,5 @@
 {
+  "root": true,
   "extends": [
     "eslint:recommended",
     "plugin:@typescript-eslint/eslint-recommended",
@@ -10,17 +11,41 @@
     "sourceType": "module"
   },
   "plugins": ["@typescript-eslint"],
+  "env": {
+    "browser": true
+  },
   "rules": {
-    "no-unexpected-multiline": "off",
-    "@typescript-eslint/indent": "off",
+    "no-prototype-builtins": "off",
+    "@typescript-eslint/ban-types": "off",
     "@typescript-eslint/explicit-function-return-type": "off",
+    "@typescript-eslint/explicit-module-boundary-types": "off",
+    "@typescript-eslint/no-explicit-any": "error",
+    "@typescript-eslint/no-empty-function": "off",
     "@typescript-eslint/no-non-null-assertion": "off",
-    "@typescript-eslint/no-use-before-define": "off",
     "@typescript-eslint/no-unused-vars": [
       "warn",
       {
         "argsIgnorePattern": "^_"
       }
     ]
-  }
+  },
+  "overrides": [
+    {
+      "files": ["rollup.config.js", "web-test-runner.config.js"],
+      "env": {
+        "node": true
+      }
+    },
+    {
+      "files": [
+        "*_test.ts",
+        "**/custom_typings/*.ts",
+        "packages/lit-ssr/src/test/integration/tests/**",
+        "packages/lit-ssr/src/lib/util/parse5-utils.ts"
+      ],
+      "rules": {
+        "@typescript-eslint/no-explicit-any": "off"
+      }
+    }
+  ]
 }