In my opinion it is best to play slow at first as speed will become a natural thing once you develop more of your left-right hand coordination.
Partially true, but not entire the story. Speed need to be trained as well, not physically but mentally. You may want to read this:
I was about to write the same thing but glad that someone who has the same experience did it first. I made a breakthrough in speed starting from lesson 4. The simple alternation exercises are excellent to practice speed bursts.
Also, I wouldn't worry too much on the damping, again, once your coordination gets better damping will become a more natural ability.
I remember someone of our classmates said he had difficulty in damping because he hadn't paid attention to it since start. It's hard to incorporate to playing later, so you'd better practice it well right now, when things are still simple. You should mentally figure out how it should be and how to do it before you actually do it, instead of just grab the guitar and let the fingers struggle. This applies to everything, not just damping.