I would also love to be able to open the popover programmatically whent he Action is set to "Command".
I have a list of supported URLs that my extension injects Scripts, which inject HTML and the Toolbar Item controls some functionality. But on non-whitelisted sites the popover can't do anything.
I struggled to find an API that could load the popover view from the SafariExtensionHandler.
Have you ever found a solution for 2: Closing popover programmatically? I have the same issue and am too unsuccessful.