I'm able to drag & drop books into my custom Status category, so it should work - though Calibre is sometimes a bit picky about how you do it, because depending a bit on how you click the book, it can start editing the entry or selecting many. But drag and drop definitively works.
I would suggest you do something like this, and set up the column with values like "reading, abandoned,finished" so you can easily drag a book to "reading" and then when done with it, either to "abandoned" or "finished".
