The plot could be that a young dragon wakes up on a snow day. It's excited to explore the winter wonderland. It tries to fly in the snow but finds it a bit difficult at first. Then it discovers a frozen lake and has fun skating on it using its claws. At the end, it returns to its cave, tired but happy, with memories of a great snow day.
A dragon's ability to fly might be affected as the cold air and snow could make it harder for it to gain altitude. Also, its fire - breathing ability might be less effective in the cold and wet snow environment. But its strong claws could be useful for digging through the snow.
If Jon Snow gets a dragon in fanfiction, he might face new challenges. People would be both in awe and envious of his newfound power. There could be plots against him not just from political rivals in Westeros but also from those who fear the power of dragons. He would have to learn how to control the dragon quickly. Given his noble nature, he might try to use the dragon's power for good, like protecting the smallfolk and rebuilding the war - torn regions.
Fanfiction about Jon Snow and the white dragon can go in many directions. Some stories might explore the origin of the white dragon and how it came to cross paths with Jon. It could be that the dragon was hidden away for centuries and Jon's arrival in a certain place awakens it. The relationship between them could also be one of mutual respect and understanding, with the dragon seeing something special in Jon that makes it want to follow him, like a kind of destiny or prophecy at work.
Daenerys Targaryen could be one to help. Since she has experience with dragons, she could give Jon Snow advice on how to care for the egg and hatch it.
One idea could be that it's a result of some ancient magic in Westeros. Maybe there are hidden spells or prophecies that allow for such a transformation. In the fanfiction, this could lead to Jon having new powers like breathing fire and flying, which would completely change the power dynamics in the story.
One interesting 'Jon Snow a dragon' fanfic could be where Jon discovers he has a hidden dragon bloodline. This could lead to him having the ability to communicate with dragons in a way no one else can. Maybe he forms a special bond with Drogon, for example.
Yes, in many fanfictions, Jon Snow is often depicted as having an ice dragon. Fanfiction writers like to explore different scenarios and powers for their favorite characters, and giving Jon Snow an ice dragon adds an exciting and powerful element to his character.