Hagrid's Umbrella

[Fer Mendoza]

No: HPFGUIDX 47414

When Hagrid got expelled from Hogwarts, they broke his wand. Now he 
hides this broken wand inside his umbrella because technically we 
wasn't supposed to do magic. But in the Sorcerer's Stone he did a 
couple of spells when he went to fetch Harry (he lit a fire, placed a 
tail on dudley and made the boat paddle on it's own).  Now I'm 
wondering, does this mean...

... if a wizard or a witch is not formally trained in a wizarding 
school, he/she  may not do magic (like Hagrid, he was banned from 
doing any more magic) and he/she is forbidden to use (or buy) a 
wand?  

...if someone refuses to go to any of the schools of magic, he/she 
may not practice or learn magic on his/her own? (Hermione seemed to 
get learned a few simple spells on her own)

...if a wizard/witch is expelled from a certain schoolof witchcraft 
and wizardry, he/she cannot enroll in any other school?

Also, who monitors those who are banned from using magic? Why didn't 
anyone send an owl to Hagrid reminding him he wasn'tallowed to use 
magic when he casted all those spells?

A sickle for your thoughts?

- Fer