Display block after some action (saved node)
You can use Rules + Rules Bonus Pack for this. The Rules Bonus Pack module has actions for enabling and disabling blocks upon a certain event (e.g. save a node). See this screencast for an explanation.

Note; Rules Bonus Pack is a testing ground for new functionality for the Rules module, so test thoroughly. However, the linked screencast above was published in 2011, so perhaps the code had some test mileage already.