Heavily agreed on your first point. I don't think typing speed so much causes productivity - more like typing speed is an indicator of experience and interest in the subject matter.

Odds are, if you are actually interested in programming, you've spent an ungodly number of hours at some point in your life in front of a machine. You spend enough time in front of a keyboard and you will get really fast at typing.

And odds are, if you've never spent enough time in front of a machine to get fast at it, you haven't really had much coding experience.