Wand Snapping is a combination punishment and symbolic expulsion from the wizarding world of the Snappee.
Only Dumbledore's influence likely saved Hagrid from further punishment and likely Dumbledore would need to say he was going to keep an eye on Hagrid to make sure he didn't murder any more people.
Wand Snapping should have also occurred for Prisoners sentenced to life because they are never going to use it again, but Rule of Cool/other Writing Cliches affected that.

 

After Harry receives the letter that says this, he gets another from Arthur Weasely (I think, might be Dumbledore, don't really remember) telling him not to surrender his wand. Now, from this I sort of get the impression that Harry has the choice to not give up his wand. I mean, he's already in trouble for the underaged magic, why would Dumbledore/Mr. Weasely want to add some sort of "Failure to relinquish wand" charge on top of that?

Say I have some drugs somewhere in my car. If a police officer thinks I have some, he'll probably say something along the lines of "Do you mind if I take a look in your car?" People are generally intimidated by police officers, and will typically say yes immediately, even if the drugs are in the car. I DON'T HAVE TO SAY YES. I have every right to not give consent to the officer searching my car, though the officer isn't required to tell me that I can choose whether he searches or not.

From the point where Harry is told he shouldn't surrender his wand, I imagined this was that sort of scenario. The Ministry can ask him to give up his wand, but he has the right to say no. He might not necessarily know he has that right, and the Ministry representatives don't have to tell him that he has the right to keep his wand.

Sorry if I rambled on, I'm running on four hours of sleep and have been awake for about 18.

 

I mean, the worst that could [reasonably] happen is that they...forcibly take his wand and snap it. What's he got to lose?

Then again they were seriously considering pressing criminal charges, so...

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I'm sure everybody has answered that being expelled has your wand snapped. As for the case with Bellatrix's wand, I do not believe it was ever implicitly or explicitly said that this was the wand she had in her Hogwarts years, as she may have gotten a new one when Voldemort broke her out of Azkaban.

But, if she did have the wand before her imprisonment, my belief is that the following could have happened;

1). Because of Ministry arrogance, they did not snap the wand as she would be in for life, and they did not believe that she could break out, or be broken out, evident by the surprise everyone showed when they learnt that Sirius Black had escaped out of Azkaban.

2). A Death Eater in power, or a Death Eater sympathizer, saved her and others wands from being snapped and saved it until they were broken out.

Of the above two, I find the former more compelling, but the answer is for you to decide.

 

As far as Bellatrix's wand goes, I could see the Malfoys arranging to claim it. Narcissa is her sister after all, so it wouldn't be that hard to argue that she wants it for purely sentimental reasons. And compared to the influence needed to stay out of Azkaban themselves, getting Bellatrix's wand would be small potatoes.