1. Is pouffy a person or a hairstyle?
Neither. It's an adjective. The fact that there is an adverb ( 'impossibly') before it should tell you that it's an adjective.
2. Why use 'do' after 'pouffy'?
The word 'do' is a noun in this context.
A 'hairdo' is a hair style (meaning the way that someone has 'done' their hair). If the context is clear, this can be shortened to just 'do'. For example, an 'up do', is a type of style where a girl has arranged her hair up onto the top or back of her head.
The noun phrase 'some impossibly pouffy do' means 'some kind of impossibly pouffy hairstyle'. This noun phrase refers to the previous noun 'her hair', and the construction is 'make [noun 1] into [noun 2]'. The writer suspects that the other girl got up at 5 o'clock in the morning and has been creating her hairstyle since then.
I'd understand the word 'pouffy' to mean over-styled: we'd imagine that the girl has used too many products and too much back-combing and blow-drying, creating a 'big hair' effect. Think cotton candy ( candy floss).
Can the word 'do' be omitted?
No.