Detect missing Syncthing, rewrite README with architecture diagram

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>

internal/tray/tray.go

@@ -14,7 +14,7 @@ import (
 	st "git.davoryn.de/calic/syncwarden/internal/syncthing"
 )
 
-const version = "0.1.0"
+const version = "0.2.0"
 
 // App manages the tray icon and Syncthing monitoring.
 type App struct {
@@ -37,6 +37,8 @@ type App struct {
 	conflictItem *systray.MenuItem
 	folderItems []*systray.MenuItem
 	recentItems []*systray.MenuItem
+
+	syncthingMissing bool
 }
 
 // Run starts the tray application (blocking).
@@ -59,10 +61,20 @@ func (a *App) onReady() {
 
 	a.client = st.NewClient(a.cfg.BaseURL(), a.cfg.SyncthingAPIKey)
 
+	// Check if Syncthing is installed
+	if !st.IsInstalled() {
+		a.syncthingMissing = true
+		log.Println("Syncthing not found on this system")
+	}
+
 	// Set initial icon
 	a.setState(icons.StateDisconnected)
 	systray.SetTitle("SyncWarden")
-	systray.SetTooltip("SyncWarden: connecting...")
+	if a.syncthingMissing {
+		systray.SetTooltip("SyncWarden: Syncthing not found")
+	} else {
+		systray.SetTooltip("SyncWarden: connecting...")
+	}
 
 	// Right-click shows menu
 	systray.SetOnRClick(func(menu systray.IMenu) {