refactor: split remote relay into own module, add lifecycle reconnect

- Move handle_remote_relay from remote_input.rs to remote_relay.rs
- Android: lifecycle-aware WebSocket reconnection on app resume
- Cleaner module boundaries between xdotool input and browser relay

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

Android/app/src/main/java/com/archipelago/app/ui/screens/RemoteInputScreen.kt

@@ -15,7 +15,6 @@ import androidx.compose.foundation.layout.windowInsetsPadding
 import androidx.compose.foundation.shape.CircleShape
 import androidx.compose.runtime.Composable
 import androidx.compose.runtime.DisposableEffect
-import androidx.compose.runtime.LaunchedEffect
 import androidx.compose.runtime.collectAsState
 import androidx.compose.runtime.getValue
 import androidx.compose.runtime.mutableStateOf
@@ -28,6 +27,9 @@ import androidx.compose.ui.draw.clip
 import androidx.compose.ui.graphics.Color
 import androidx.compose.ui.platform.LocalConfiguration
 import androidx.compose.ui.platform.LocalContext
+import androidx.compose.ui.platform.LocalLifecycleOwner
+import androidx.lifecycle.Lifecycle
+import androidx.lifecycle.LifecycleEventObserver
 import androidx.compose.ui.unit.dp
 import com.archipelago.app.data.ServerPreferences
 import com.archipelago.app.network.ConnectionState
@@ -60,10 +62,32 @@ fun RemoteInputScreen(onBack: () -> Unit) {
 
     val ws = remember { InputWebSocket(scope) }
     val connectionState by ws.state.collectAsState()
+    val lifecycleOwner = LocalLifecycleOwner.current
 
     BackHandler { onBack() }
-    DisposableEffect(Unit) { onDispose { ws.disconnect() } }
-    LaunchedEffect(activeServer) { activeServer?.let { ws.connect(it.toUrl(), it.password) } }
+
+    // Connect on server change + reconnect when app resumes from background
+    DisposableEffect(lifecycleOwner, activeServer) {
+        val server = activeServer
+        if (server != null) {
+            ws.connect(server.toUrl(), server.password)
+        }
+
+        val observer = LifecycleEventObserver { _, event ->
+            if (event == Lifecycle.Event.ON_RESUME && server != null) {
+                val state = ws.state.value
+                if (state != ConnectionState.CONNECTED && state != ConnectionState.CONNECTING) {
+                    ws.connect(server.toUrl(), server.password)
+                }
+            }
+        }
+        lifecycleOwner.lifecycle.addObserver(observer)
+
+        onDispose {
+            lifecycleOwner.lifecycle.removeObserver(observer)
+            ws.disconnect()
+        }
+    }
 
     Box(
         Modifier