"I don't understand how memorizing where D is, allows you to know where all of the othe notes are on the fretboard without using rote memory for all of them?"

This tutorial is simply an introduction to the musical alphabet.  Anders is only introducing you to the musical alphabet and how it works on the guitar fretboard in one location, that set of notes in the tablature. 

"Also,  in the example given for the tutorial (attached), the G note is found on more than one string, at more than one fret position?"

Yes, all notes of the musical alphabet repeat over & again all the way across the guitar (and every musical instrument).  But keep in mind Anders is only introducing you to this idea.  Even when he gets to the final lesson of that tutorial, he's just introducing you to that idea.  You aren't expected to know them all right away.

Keep in mind there are only 7 letters A through G, and 5 accidentals in between some of them.  That a total of 12 notes that repeat over & over again.  So, it seems like a lot when you are just getting started & every thing is overwhelming!  But it's actually a small list & you can learn it a little at a time.

If you are interested in learning more about the pattern the musical alphabet makes on the fretboard you might find this set of lessons helpful.

https://www.guitartricks.com/lesson/11078

I also have this individual lesson that shows a method for gradually learning the pattern musical alphabet makes on the fretboard.

https://www.guitartricks.com/lesson/11526

Hope that helps!