If I lose battery and the 3DS dies, is the save lost?
If I lose battery and the 3DS dies, is the save lost?
Progress, likely; the save itself, not sure. I had a freeze once and lost a few hours of progress, but the save didn't get corrupted or anything, although that doesn't mean that it couldn't happen. Make dure to do freaquent periodic backups of your saves.
As for the power button, I believe it's best to wait a little bit after first pressing the power button, just in case, before clicking "power off".
Realistically, the save would only be lost/corrupted if the 3DS lost power in the narrow window immediately after hitting Home/Power when the save is being written.