I don't want to do anything unusual with the pegs. Could I make them out
of
3/4" stock?

Yes.

is black walnut o.k. for turning?

Yes, but sand thoroughly or tool maks will br pronounced when you add
finish. Plan to make a few extra pegs and toss the ones that didn't come out
so well.

how much longer than the
finished peg should I cut them?

Here's the fancy way:

* Make the tennon 1" long. Initially it will protrude 1/4" past the back of
the workpiece
* Drill all the eay through your 3/4" stock.
* Make a handsaw or bandsaw (narrow) kerf anout 7/8" deep, at the end of the
tennon.
* Cut a wedge of wood a little wider than your kerf
* Insert tennon with glue into hole.
* Put a little glue on your wedge and pound it in the kerf (effeectively
spreading the back of your tennon).
* Use a hand saw to trim the protruding tennon (you can leave a little bit)
* Trim flush with a sharp chisel

Should I just tap them in (to holes drilled in back piece) with some wood
glue?

You could.

maybe reinforce them with small screws from the back?

No, because screwing into endgrain is not reccomneded. IMO you would benefit
more from a deeper (though) mortise.

One last question. How would you attach this piece to the wall? I am
thinking that perhaps I will drill countersunk holes in the back piece
and
screw it into the studs, then cap the holes with little raised and rounded
plugs.

IMO it would look better of you could find some mounting hardware that can
be buried in the back of this thing.