To edit products and set which categories they are assigned to in mass, you can use the Product Mass Editor.
Essentially what it does is allow you to quickly edit many products at once.
To use the Product Mass Editor, navigate to Catalog > Power Features.
Next, click on Product Mass Editor.
You can edit more columns than are shown by default here by clicking Choose Columns To Edit at the top of the page.
You can also Edit the current layout, create a new one, or save it for later.
Either way, this will open an option screen that allows you to display a variety of options on the Product Mass Editor section. Once you have checked the appropriate boxes, click “Apply” in the top, right corner of the screen.
You can then hit the "Edit Row" icon and start entering in your info.
Once you are done editing, you can move to another product or click save.
If you select "Edit" it will bring you to the product editor for more options.
By default, you can change categories but editing them through the product mass editor is not enabled and to change their structure you will need to do so in the Catalog.
In the above example, there are multiple sub-categories with the same name in different parent categories. To edit this info you need to edit the product with the full product editor and select "Categories"