feat: embed frontend into backend server with full-stack build pipeline (RCA-20)

The backend now serves the Vue admin UI as static files with SPA fallback,
eliminating the need for a separate web server. Dockerfile builds both
frontend and backend in a multi-stage pipeline. Added build:web and
build:all scripts, updated CI to verify frontend builds, and fixed
vitest config to exclude Playwright tests.

Co-Authored-By: Paperclip <noreply@paperclip.ing>

.gitignore

@@ -2,6 +2,7 @@ web/node_modules/
 web/dist/
 node_modules/
 dist/
+public/
 data/
 *.db
 *.db-wal