Hi Super , I'm so sorry you & your DD are stil going through all this:( but I have to say, the more you write, the more I'm thinking OMG are you me
my own DD has had to deal with all the symptoms you describe, again often for weeks on end & I too also have a M.E. diagnosis, though I've bender felt that right if I'm honest - so I can totally understand your problems & mistrust of Doctors, it's not an easy diagnosis to be heard with :( & really sorry your hospital s so bad & you don't trust them either - do you have the medical walk in centres near you yet ?? - I'd find out if I were you as that might be a better option - our A&E Told me they are able to queue jump referrals to the hospital & itscabiynlike an A&E for more minor non life's threatening stuff.
10 years down the line I am finally getting some answers, my DDs Hypermobility Syndrome Diagnosis has led to my being re referred to the Rheumatologist again, it's hereditary & my reading up to help my DD I can see why they are referring me - it is linked to both ME & Fibromyalgia, symptoms cross over massively & like my DD links my other problems all into one too - I also didn't consider myself very flexible, it was all just normal for me & I'm no contortionist, DD is a bit bendy, but again, no big deal, we both always sitbin the W form on thevfloor though - i had no idea that wsnt normal until recently - but she still scored highly on the diagnostic Beighton Score - I score highly on the Brighton score - I would recommend you look into those yourself, just in case.
The other thing that springs to mind with your DD though is Migraine - in kids migraine manifests in the stomach the flashing lights, light/noise sensitivity, nausea etc etc, are all symptoms of that too, everything you mention could be migraine - I can't link on the iPad but look up Migraine.org - it's actually run by the migraine nuro from our local hospital & there's tons of good info on there.
The other thing I wanted to say - with your DDs intolerances - my own DD is exactly the same - if she's well she can tolerate wheat & dairy without any problems, though generally not lactose, she needs cheese to be cooked or mature as is lower in lactose, she needs lactose free milk etc - she can eat spelt flour fine, so I can cook pizza, pies, cakes etc with that, or other flours
you don't mention additives though, & I wonder if this could be a problem for your DD as it is mine?? These can be a far bigger trigger for my DDs stomach & skin problems than wheat & diary & kids medicines tend to be full of artificial sweeteners, colours & preservatives -
My DD has to have the white adult parecetamol or Ibruprophen as Calpol will make her really ill - kids antibiotics are also packed full of additives, these affect my DD so badly that she has ended up in A&E & they got as far as hooking her up to a drip for surgery, telling me it was appendicitis - a more senior surgeon said it was actuallyncolitis, ie inflammation in her colon - caused by the additives in the medicine, on top of the bad side effects of the antibiotics themselves - her skin rashes with these additives too, but they also affect her insides, causing stomach pain & diarreah that led to her original IBS diagnosis.
If my DD needs antibiotics, she has to have the adult capsules, which we have to break open & mix the powder with honey, to avoid the red die & shellac in the capsule it's self - this helps a lot to avoid making her stomach worse.
We also give her Viridian stomach flora powder, mixed into a smoothly or something - this helps rebalance her gut & again helps her get well - the spirulina we give her gives her iron & most vist & minerals in good doses along with an immune boosting enzyme.
When her stomach is bad my DD tends to get pain & diarreah, with nausea & dizziness & exhaustion, aches & pains - keeping her off wheat & diary for a few days & giving her white bulking foods such as rice & potatoes & I find turmeric can work wonders - it's a curry spice, but it's used medicinally in India too - it soothes & reduces inflammation in the guts & has been a godsend for us - I usually just add a teaspoon to the water for cooking the rice, DD eating turmeric rice a few timesba day, with no wheat, disry or additives usually helps settle things pretty quickly :) & we use heat pads to soothe her stomach too.
As far as school goes, I agree with the others, if she's vomiting or diarreah, then school rules with ean they will want her to stay home - though with stress related chronic diarreah, the school should work with you, ours do with DD - she wants to be at school, so they accommodate that by letting her rest in the quiet corner if she needs too, my DD can really struggle with class noise too
I would highly recommend making an appointment to see the school nurse, ours was a huge help with both getting the school to take it seriously & advising on which specialists to see & even just ideas on possible cause. It could help you stop worrying about keeping her home & help with strategies for combating the anxiety over sickness at school.
As a foot note & please don't take this as criticism - I'm speaking from personal experience of being ill & exhausted myself & dealing with extremely ill DD & finding it hard to get help for years - cut yourself some slack & understand & that your exhaustion & the stress you are dealing with will make you highly anxious, it in turn will make you mistrusting of your GP & it will make it hard for you to listen & understand properly what they tell you - you need to speak up & not be too proud to say - I am ill & exhausted myself & finding this hard to follow - please help me understand what this is & why you are giving DD this treatment & what happens next - make notes, ask questions & write down the answers & double check with the doctor that you have understood correctly - it's too easy in our situation to misunderstand & it leaves you feeling more stressed & that you aren't been heard - been there, done that 
Good luck