Yes, of course, you can change any settings of the category you’ve copied. Moreover, you can use a “Find and Replace” option on the “Duplicate Category” page. With this functionality, you can automatically modify category name, description, etc. Follow the link to see the settings. 

Yes, you can copy all product relations by choosing parent category for the duplicate on the Duplicate Category page. Read on to see the settings.

Yes, sure, the extension offers a possibility to copy categories from one store view to another. Go on reading to see how to make it correctly.