Backstory (bullet points 1 & 3)
Lucian Aldor was in need of some way to avoid restocking her annoying-to-acquire-sword-fodder every time nosey people with sharp objects showed up at her doorstep, and she thought a baby dragon might be just the thing she needed for her experiments.
Snowball ended up in Ms. Aldor's home before hatching, but the mutant-dragon to be never found out how. Snowball always thought that the wizard was her mommy; no reason to look for mommy when you can see her every day. And anyway, thaumaturgical experimentation is how all good mommies take care of their babies, right?
Eventually, the wizard's experiments bore fruit in Snowball's fast healing and improved constitution. The magic fractured Snowball's mind, but having a more pliable test subject didn't seem like a drawback to Lucian. Not as long as it wasn't her own mind.
One of the side effects was somewhat animal-like behavior. That, combined with the dragon being the size of a large house cat, inspired the spell caster to give Snowball her name.
After trying everything she could think of, yet failing to extend Snowball's regenerative abilities beyond the dragon's body, Lucian declared the experiment a failure. No longer necessary for experimentation, Snowball was relegated to guard duty.
These days Snowball misses her mommy, but she contents herself with freezing and then consuming any meat that wanders into reach; often followed by carving/gnawing bones.
Motivations and plans (bullet points 2 & 5)
I'm not sure what normally motivates white dragons besides a safe pile of treasure to sleep on, good cold meals every day, and the frozen tears of defeated foes.
In Snowball's case, she'll also be (at least somewhat) motivated to find out what happened to her mommy. In the future she'll almost certainly discover that her real mommy is a dragon, but how she feels about that will mostly depend on how she finds out and what she learns beforehand.
As per GM Monster's question:
If she had treasure but not friends/family, I think she'd find a way to be some poor creature's mommy and then be a good mom conduct an endless a terminal series of horrible thaumaturgical experiments. She'd replace each of her unfortunate captives as they expire, possibly by eating them, and proooobably feeling bad about it yet repeating the cycle after each death anyway.
I'm not sure that she could end up with friends/family but no treasure. Whether she had treasure or not, I'm not sure if she'd want to experiment on any of her combat buddies, but she might try to adopt someone, hehehehe.
Description (bullet point 4)
Snowball's head is slightly deformed, looking rounded in front (a bit like the blue-eyes white dragon from Yu-Gi-Oh). She otherwise looks much like every other white-scaled, ice-breathing, flying lizard-of-death.
If we get more beneficial mutations later, some of her armored scales will become more like armor plating than usual for a white dragon of her age category.
