Yes, when you Set Input Game UI Only, feed the Game Menu widget Return Value reference into In Widget To Focus:
You may also need to enable isFocusable flag inside of the Game Menu widget for this to work.
When it comes to mouse visibility, you can handle it via the controller - Get Player Controller → Set Show Mouse Cursor.
Widgets, however, override this setting and force the cursor to appear by default. Each widget controls it separately, though - there is a small hidden tab in the Behaviour panel of each widget. This will allow you to individually control cursor behaviour over every single one of them:
Hope this makes sense.