Okay, I don't think we have a specific set of guidelines, but I'll try and answer the questions one at a time.
1. Logically, I don't think a letter is very likely.
Dear Linnet,
Sorry to have to tell you, but your grandmother died yesterday,
Lots of love, Dad.
I can't honestly imagine any parent writing such a letter. Far more likely that they'd write to the Head of house or even write and ask if they could come to the school and break the news to their child themselves. If it was a family member they weren't close to - an aunt who had been ill for a long time, for example, I think a letter would be more reasonable, but for a grandmarent or parent, I really can't see it happening.
2. I think in almost all circumstances the student would *want* to go home. If it is a parent, they would likely want to support the other parent if they were old enough; help with arrangements or just get away form people asking questions or worse, avoiding them because they don't know what to say.
As for exams - yes, of course they would be allowed to stay. But speaking as someone who sat an email three weeks after my father died - they're not likely to be in a good frame of mind. I had to be escorted out of the exam hall halfway through because I couldn't stop crying and needed to get a grip on myself, and I am fairly certain I wrote utter rubbish for at least one question (I had special dispensation from the doctor which was taken into account).
3. You can submit the update at the end of the school year when you next update the character sheet. I think that would be the easiest way around it.
Hope that helps a bit. Feel free to ask if anything needs clarification (oh, and there's a big section on how Hannah Abbott's mother's death affected her in her sheet if you want to take a look)
~ Carys