WordPress.org

Does this plugin require me to overwrite WordPress core files?

No, there is nothing more you have to do except uploading and activating the plugin.

Does this plugin allow me to reorder subcategories as well?

Yes it does. You'll see a 'more >' link next to the categories that have subcategories. Click on that link and you'll be able to reorder those subcategories.

The Category Order panel seems to work fine but my categories are not reordered on my blog. What's going on?

Make sure you're not specifying an order in wplistcategories() (e.g. wplistcategories('orderby=count');)