Yes, the order placed by the same account on either of the 3 platforms is accessible on other platforms as well.
To check your past orders you have to navigate to Orders Panel.
- In Planet Explorer on the side pane, you will have an option to view your order history
- In QGIS/ArcGIS once you log-in with your credentials on top you have to click on "Show/Hide Order Status Panel" which will allow you to see your order history
Orders made by API cannot be downloaded by PlanetExplorer nor by QGIS/ArcGIS Plugins.