Yes, in the original Grimm's Fairy Tales version, Snow White does die. The jealous queen orders a huntsman to kill Snow White and bring back her heart as proof. The huntsman, unable to do so, lets Snow White go into the forest where she eats the poisoned apple given by the queen in disguise and appears to die.
The original Snow White story doesn't really deal with Snow White's mother in a way that indicates she is alive. It's the stepmother who is the main antagonist. There are no details given about Snow White's mother, which makes it seem that she has passed away or is not relevant to the main events of the story.
The male Snow White in fanfiction often has different character traits. He might be more physically assertive compared to the rather passive female Snow White in the original. He could also have different goals and motives. Instead of waiting to be rescued, he might be on a quest to save others or himself in a more active way.
In the 'Disney Snow White Christmas Story', there would be Christmas elements added. While the original is more about her escape from the Evil Queen and her life with the dwarfs, this Christmas version would have things like Christmas decorations, carols, and special Christmas - themed events.
No. Snow White was in a death - like sleep after eating the poisoned apple. But the prince came and his true love's kiss broke the spell. So she was very much alive at the end of the story. She and the prince got married and started their new life together in the castle. There was joy and celebration in the kingdom as Snow White was alive and well.