[Plugin: My Category Order] Is it possible to edit the URL of certain categories?

It’s not going to be possible. The widget uses functionality built into WordPress. Your only real option is copy the HTML from your source, stick it in a Text widget and then tweak it how you want.

Or create a link category and add the links to your categories and other pages. If you install My Link Order you could specify which link category to display and get your customized links. Not sure if you can put in a link without a URL.