feat: add EditEventScreen with calendar and chat mode support

Add a unified event editor that works in two modes:
- Calendar mode: Create/edit events directly via EventService API
- Chat mode: Edit AI-proposed events before confirming them

The chat mode allows users to modify proposed events (title, time,
recurrence) and persists changes both locally and to the server.

New components: DateTimePicker, ScrollableDropdown, useDropdownPosition
New API: PUT /api/chat/messages/:messageId/proposal

apps/client/src/hooks/useDropdownPosition.ts

@@ -0,0 +1,37 @@
+import { useCallback, useRef, useState } from "react";
+import { View } from "react-native";
+
+type DropdownPosition = {
+  top: number;
+  left: number;
+  width: number;
+};
+
+/**
+ * Hook for managing dropdown position measurement and visibility.
+ * @param widthMultiplier - Multiply the measured width (default: 1)
+ */
+export const useDropdownPosition = (widthMultiplier = 1) => {
+  const [visible, setVisible] = useState(false);
+  const [position, setPosition] = useState<DropdownPosition>({
+    top: 0,
+    left: 0,
+    width: 0,
+  });
+  const ref = useRef<View>(null);
+
+  const open = useCallback(() => {
+    ref.current?.measureInWindow((x, y, width, height) => {
+      setPosition({
+        top: y + height,
+        left: x,
+        width: width * widthMultiplier,
+      });
+      setVisible(true);
+    });
+  }, [widthMultiplier]);
+
+  const close = useCallback(() => setVisible(false), []);
+
+  return { ref, visible, position, open, close };
+};