fix: commit pre-compiled dist and simplify sync Dockerfile

Server npm proxy silently drops devDependencies, making TypeScript
unavailable in Docker. Solution: compile locally and commit dist/.
Dockerfile now only needs prod deps + Playwright, both of which
install cleanly via the public registry.

Also fix TS2688/TS2304 errors: add DOM to tsconfig lib and cast
querySelectorAll results to Element inside $$eval callbacks.

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

sync/dist/types.js

@@ -0,0 +1,2 @@
+"use strict";
+Object.defineProperty(exports, "__esModule", { value: true });