Parenting my 13yr old DD is a thankless task these days

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I'm really struggling with my DD. She is 13 and in year 9, august born.
She is vile at the moment, we don't understand why. It's in the last 3/4 months, she has changed so much. Nothing bad has happened to her as far as we know.
She calls me names, hits me sometimes, does the very bare minimum homework. Her school books are very untidy and messy with empty pages - I don't know how she gets away with it at school, but she is never had a detention for it and this is a school that gives out a lot of detentions.
Her room is extremely messy.
Her messiness has been an issue for years but her rudeness is recent.
Every morning is a battle with her as she is extremely stubborn and has aversions to certain things with no logical reason. This morning she insisted she cycle to school as usual even though the snow was a few inches deep, we are less than 30min walk from school. In the end after arguaing for ages we had to threaten to lock her bike up and take the key to prevent her from cycling. She has no issue with general walking and is actually a very sporty child, but she just digs her heels in at every opportunity with no good reason. This is just an example but indicative of ridiculous things she won't budge on.
She is a very fussy eater too, refuses to eat certain food even though she's never tried them ever - just refuses to even try.
She is grumpy and rude all day, to us and to even close family.

I just can't take it anymore. DH is too soft with her and won't discipline her properly. I have DS too who's 15, he tried to engage with her and play with her, but a lot of the time she is rude to him and refuses to, although sometimes if she is in a good mood she will.

I'm worried about her general demeanour, her school work, her attitude to us, and it's getting me down.

Is this just normal teen girl behaviour is this something more we need to look into?
Thanks for reading.

Sounds a lot like DD. She has a special interest in rock climbing and we do everything to facilitate it, including giving up our Friday evenings to sit in cold drafty climbing centres every week and DH goes climbing with her during school holidays.
Would she benefit from a formal diagnosis or do we need to just find ways to manage this I wonder?
She's been a bit better in the last few days. I feel we have indulged her and been a bit too soft on her behaviour and recently I have backed off a bit and DH has much sterner with her when she is rude and she has been a bit less grumpy and hanging out with us more. Although she is still "difficult" .

She is actually great at routine changes and transitions. She takes things in her stride and can respond to instructions well under pressure. She is a good person to take on a stressful, unpredictable holiday for instance as we've found out on many occasions! She just goes along with anything. But at home and to a degree at school is she is very stubborn.