I'd definitely avoid 'slovenly'. This describes an entire appearance and behaviour, suggesting that someone is dirty and lazy. It's most often used as an insulting description of a woman and her lack of care of herself and her home. This person's hair isn't tangled, either. Tangled means that it's knotted. Unkempt, messy or untidy would fit. But, of course, this is a carefully arranged 'look' from a salon - the messiness is deliberate. A nice word to describe deliberately untidy hair like this, on a man or a woman, is 'tousled'. Try googling it and you'll see what I mean. Otherwise, there's a look called 'bed head'. If someone wants to look like they've just got out of bed, this is what they call the image.

All of your words make sense. There is a quite funny idiom which is sometimes used for situations like this where you look a bit messy, which doesn't just apply to hair, but your appearance in general. You can say you look like you've been 'dragged through a hedge backwards.'

Tangled, unkempt and messy all make perfect sense, slovenly might be right but someone would probably ask you for clarification!!