Having worked for several years with 'difficult' children, especially teenage girls I offer the following:
Are there other problems other than just those you outline?
The move from the previous school could have affected her, maybe she isn't settling in too well.
Is she doing well otherwise with her work and classmates?
Often this behaviour manifests due to other issues such as bullying by other students or teachers.
It is always best to treat these issues as quickly as possible because no one wants her school work to suffer, and there may be intervention available in school.
Despite the government claiming they have spent more money on mental helth in schools this money may not have trickled down to reach young women like your granddaughter,
The incidents you describe could be symptoms of autism, but diagnosis can be extremely slow or very expensive. Some private assessments will not be recognised by the LA so try to work with the school as much as possible,
Maybe she needs a reduced timetable for a while or a referral to CAMHS, but again the waiting lists are very long. Some students wait years.
You say she was suspended from school- maybe the school has different provision for school transfers? If you aren't careful the new school may try to send her back- this is not unusual.
12 years is still young- my guess is there is a lot going on under the surface here.
Having said all this you are a grandparent, not a parent so your role in her education is limited if not irrelevent. I would work closely in a supportive way with the parents and just be there for your grandaughter in other aspects of her life.
Raise her self-esteem, praise her for what she gets right but don't bribe her into good behaviour with buying her stuff.
After working with girls like this for many years it isn't a simple picture, often it is far more than simply the school having 'silly' rules and I am afraid it sounds as if you are in for a protracted time of trying to guide this young woman through some difficult situations.
Try not to focus on her education too much, just be her granny who loves her for what she is and find ways to have fun in other activities that mean you all get to relax a little,
I wish you well.