how the hell do you people even manage to constantly snap these things
It depends...And I'm assuming it's snapped at the hinges rather than down the middle.
Does it still turn on? Does it still function perfectly other than the screen being detached? Do both screens still 'work'? Do games still play? If it's only the plastic hinges that have snapped and the DS still functions fine otherwise, I don't see why you shouldn't be able to transfer data to a new 3DS.
If the thing is completely wrecked and is useful as nothing more than a paperweight...I'm going to say no.
NOt yet... But soon...
It's not yet snapped? Obviously, just try and be careful with it then and you should be fine. If it's anything like the old DS then aside from the hinges, the top screen is only really connected by a few small wires that could easily be broken once the hinges do break.
Either way, as long as you can still turn it on, play games etc. you should be fine.