Woodturners use super glue for a number of things, including a great shiny finish on turned bowls. It is frequently used to repair cracks in wooden bowls, but it does not hold up to the stresses of thin wood expanding and shrinking with humidity.
As for staining, staining first is ok if you are not repairing something already stained and finished, or you are completely stripping and re finishing. Glue to fill gaps does not take stain and is noticeable. Where the thin glue can seep into a hair line crack fine. Even the thick stuff is no good at gluing where there are voids. I had great success fixing chunks tore out of walnut stocks by matching the grain from another piece of walnut and cutting the hole in the stock and the matched piece to fit nearly exactly, and then use epoxy, not super glue. I even glued a piece to the end of a trapdoor stock where somebody cut off the fore end. cut the old stock oblique and then selected a piece with matching grain, turned the grain the same direction and cut matching piece the same oblique angle and then inlet the new piece for the barrel and nose cap. I was able to finish the original stock and new piece (about 8 inches) such that even I had trouble spotting the repair. Super glue would never have held up for such repairs.
Wood turners often glue smaller contrasting blocks of wood into a larger piece for turning that leaves a fancy design. They use standard wood glue for gluing the segments because super glue does not hold up as well to holding the blocks together during the stress of turning.
Also, use care when using superglue, it gives off noxious fumes just by itself. Wood turners often use it in connection with a hardener and the fumes from that reaction can cause chemical burns to the nostrils.
