Add Syncthing installation detection (PATH + config file check) to both
the tray app and setup installer. When missing, the tray shows an
"Install Syncthing..." menu item and the setup opens the download page.
Rewrite README with Mermaid topology graph, per-binary dependency tables,
project layout, API endpoint reference, and shields.io badges.
Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>
