The good news is that we can also show internal iLogic Forms in a DockableWindow like that, not just 'global' ones. The bad news, as I suspected, is that the DockableWindows we create that way (with an iLogic rule) will not survive restarting Inventor. So, you would just have to create them again after restarting Inventor.
Also, keep in mind that when you launch a global iLogic form, it will be focused on whichever document was 'active' at that moment, and it will not change its focus to another document, just because you switched to viewing another document. In order to get the global iLogic form to focus on another document, you must first close that form, then open it again while the other document is active (viewing it on your screen).
So, you may want to setup something like this to use the Event Triggers, if trying to keep it simple, and not use an add-in. When a document opens, if it is the 'active' document, use a rule to ensure that the window exists, then either launch and add the form to the window, or refresh the form within the window by Clearing the window, then adding the form back to the window again. When a document closes, if it is the 'active' document, use a rule to Clear the window. If you clear the window when closing the document, then you should not need to clear it when opening another document.
This may be complicated to do using the regular Event Triggers dialog though, because documents can open and close invisibly in the background, by simply opening assemblies or drawings that reference them, or opening and closing them invisibly by code, and that will trigger those events. You would only want these window and form changes to happen for the document that you are actively viewing, not for invisible background documents. That is why you will most likely want to use 'ThisDoc.Document' to identify the document that triggered the rule to run, but then you will likely want to check it against 'ThisApplication.ActiveDocument' to see if they are the same. And if they are, then move forward with the rest of the rule, otherwise exit the rule, without doing anything else.
I attached a few text files for similar external iLogic rules. But these are just the basics of what would be needed to set up a full automation project that will maintain, and update a form like this across all documents that might open and close while Inventor is open, using just iLogic rules and the Event Triggers dialog. This type of setup is best handled by an add-in.
Wesley Crihfield

(Not an Autodesk Employee)