TL;DR - 1a) Do we know of any other students that were actually expelled?
TL;DR - 1b) What constitutes a complete education? Do all children have to pass OWLS and NEWTS?
The only example I can remember is: Hagrid. He was fully expelled, wand broken and everything.
It is also my recollection that the wand is broken because without a completed education the wizard will end up being a danger to the public by not performing spells properly.
However we see the following:
- Fred and George left the school voluntarily so there was no official to confiscate and break their wands. They end up opening their joke shop.
- Same goes with Harry and Ron so they can pursue the Horcruxes. They end up working as Aurors directly after the war.
- Hermione leaves but goes back and completes her education.
This leaves the following questions:
- Since Fred and George did not finish, why did the Ministry not show up at the shop to confiscate and break their wands?
- Same with Harry and Ron. Why when they went to work for the Ministry was nothing said or done?
Harry and Ron are the HEROS so I am guessing that no one would want to enforce the law in their cases. Kingsley actually recruited them to head up the new Auror department.
Fred and George, however, leave before the end of the war in a time where the Ministry is corrupt. So it would stand to reason that they would be targets.
We also know that many children are home-schooled throughout the world. So there must be some standard that the Ministry would deem acceptable to not go break the wands of all the wizards that are home-schooled.
It almost seems to that this is just a setup for Hagrid's character and role, plus a background threat to Harry whenever it convenient for the plot...
Edit to account for Fantastic Beasts and Where to Find Them:
We know that Newt Scamander was expelled from Hogwarts, but he definitely still has his wand!