1) There are many ways to handle scenarios like this. One common way is to put your state in a "manager" class, then pass that class to both panels when you create them. Another more advanced way is to use events. Panel1 would fire events that panel2 handles, and panel2 fires events that panel1 handles.
2) No, there is no cursor hover. There are 2 things you can do to work around this limitation. If you just want to show some text for a node/edge then put the text in the node/edge table and use a passthrough mapping for the tooltip visual property. The other thing you can do is add menu items to the right-click menu for nodes/edges.
3) This is a known bug. There is no way to show a dialog or popup while running a task. You can show a dialog inside a call to SwingUtilities.invokeLater(), but the dialog will show up after the task completes. If you are showing a dialog because the task needs input from the user the standard way to do that is to use fields with @Tunable annotations in the Task.