A: Is the META Keywords Tag still important?

Technically, you're both incorrect. While not utilized by search engines anywhere near the extent it used to be, the META Keywords Tag is still indexed by a couple of search engines and so is worth including within a site's HTML code if only to provide those search engines with as much information as possible about site content so they can index it correctly.

But it is certainly not an important tag and you can safely exclude it from your code without concern. You'll find which engines still support the META Keywords Tag here.