What do we call the thing [slid in eyes to apply khol] lying with the kohl bottle in the picture?
It's simply called a "kohl applicator".
Applying kohl on the waterline involves dipping a rod-like applicator into kohl powder and then, after flicking off the excess, passing the rod through the base of the lashes while the eyes are closed. If done correctly, the wet conjunctiva picks up the powder from the side of the rod creating a smooth, even line.
Since it has no specific name, you can call it any of the words you mentioned. Although twig and scion can be taken as something much thinner. I'd prefer "rod" or "stick".
As mentioned by @xeesid in the comments, "kohl needle" could be used too.