Weapon of Choice: There's not a whole lot of objectivity involved with phrases like "somewhat of a sword short that can be lethal in skilled hands." Considering the massive variety of enemies in the current iteration of Vale, what a weapon can and can't easily kill is pretty objective. And, to be honest, there isn't much of a need to describe a weapon's effectiveness.
Spells: I saw that you mentioned that you needed help with your spells, and after seeing them myself, I'm inclined to agree. Though, currently, it seems there won't be much benefit from my analysis, I'll go ahead and explain should a spot open up later.
Crush: There's really not much need for the "and Lexi traps her foes in a rapid vibration sphere." Unfortunately, it convolutes the meaning of the spell, along with just making things confusing.
Accelerate: Honestly, this is just a speed buff that could've been incorporated into Glide, so I'd remove it and just mention that Glide increases Lexi's speed as well. Oh, and the physically draining thing is a bit of an issue as well, considering the PvE environment. After all, it's only a speed buff.
Glide: I don't quite think you mean "solidifies the air," since that would basically be creating a solid platform of air, whereas you describe something akin to traditional gliding.
Shield: Not really bad in concept, in fact, I really like the idea. However, should something of greater speed or something too heavy interact with the barrier, it would be unable to defend Lexi from the attack, since it sounds like the barrier is simply deflecting attacks.
Appearance: Not much to say here, other than that you should mention Lexi dyed her hair white, since white hair is unnatural and Vale is set in the modern world. There are certain spells that could potentially tint a mage's hair, but they are few and far between.
Personality: Only real issue I have is a continuity problem. You mentioned that she is kind and caring, but at the same she is very emotional and quick-tempered. In other words, it'd be next to impossible for her to display this kindness, since she's so prone to blowing up on people.
History: There are two main issues that I have here: the first being this unmentioned shyness that seems to pervade Lexi's history. The second issue is her magical experience. Before being accepted, she's only had a freak run in with a bear to trigger her magic and some time "play[ing] around" with magic. She needs to have at least proactively tried to develop her skills for Vale to be interested in her.