My seamstress qualifications are roughly nil, but from the gliders I've
flown in I'd suggest:

1) What material are pockets typically made of? Anything special that
I won't find at the local sewing shop?

Pretty much anything you like, from leather to orange velour.
Leatherette/vinyl seems popular.

2) What is used at the top of the pocket to make it stretchy.

Elastic. Sew the hem of pocket front with the elastic inside, stretch
elastic, sew elastic to the seams at either end of the pocket (this is
from my mother's method of making elasticated waist skirts for my
sisters back in the '60s - then it was known as "knicker elastic").

3) The pocket is attached to a long piece of matching material along
the interior. I assume that this long piece has some stiff material
inside to give it rigidity. What is it?

Anything from cardboard to spring steel. I'd suggest a piece of thin
plastic (ice-cream tub?).

4) I assume that the old pocket is glued to the wall of the glider.
Any tricks in removing this?

Brute force I think, though maybe a sharp knife eased along the join
would help.

5) What do I use to attach the new pocket to the wall of the glider?

Glue velcro to side wall and pocket - that way you can make new pockets
when these wear out. Latex-based glue (brand name Copydex in the UK) is
good for fabric to fabric joins.