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

docs-src/_data/api.11tydata.js

@@ -1,7 +1,9 @@
 const fs = require('fs');
 
 module.exports = () => {
-  const customElements = JSON.parse(fs.readFileSync('custom-elements.json', 'utf-8'));
+  const customElements = JSON.parse(
+    fs.readFileSync('custom-elements.json', 'utf-8')
+  );
   return {
     customElements,
   };