Are their all the different types of wands like for instance, mine: sycamore wood, unicorn hair, 10 3/4 in, unyielding flexibility? Or are they the wands of all the different characters?

They have very limited types of wands. They have replicas of known wizards as well as original Ollivander wands. Some of the Pottermore wands are in the shop: Birch, Rowen, Ash, Alder, Willow, Hawthorne, Oak, Holly, Hazel, Vine, Ivy, Reed and Elder. For the core of the wand, the wand keeper is the only person to tell you the core. Unfortunately, Sycamore isn't one of the available options

They have a limited variety. The do not have the Pottermore wands.