LOL Boy does this bring back the memories -- bad, BAD memories!
When my oldest daughter was 10 we went through this. And I didn't know what was wrong with her! Because in my generation the worst hormones hit at about age 13 or 14. I remember we were on a family trip and stopped in to visit my best friend from high school who had daughters older than mine. And thankfully she explained it to me, that girls hit those hormones the heaviest just before puberty kicks in, and now days the process starts so much earlier. (Some scientists think it may be all the hormones we're consuming in our milk and meat, but this isn't proven yet.)
Anway, having gone through this with two daughters and talked to lots of other moms, my best advice is to remember how you coped with the terrible twos stage, the toddler tantrums. Set firm limits over the things that matter, keep your cool and your sense of humor, and remember above all: it's not about you!
So just because she falls apart about everything you say or do... remember its not that you did anything wrong. Just let it roll by. Try to make sure she has a quiet, private space to call her own and that she understands she can go there when everyone seems too frustrating.
Be sure she understands that its okay that she has her melt down moments. She just needs to learn appropriate ways to handle them. (Like going in her room and beating a pillow against the bed. Or having a good cry if she needs it, etc.)
This is also a great time to encourage her in exercising. Something fun like swimming, bike riding, nature hikes -- whatever she might be into. Because this will help detox her body and give a physical outlet to her pent up emotional energy.
Make sure she drinks plenty of water. Being dehydrated tends to make us all feel tired, cranky, and short tempered. Plus her body is changing and is needing more fluids than she is used to drinking. Trying to cut down on sugars can ease mood swings (up to down) as well.
And of course this is a good time to develop other self nurturing and self calming strategies -- asking for a hug, curling up with mom to watch a good movie, soaking in the tub, etc. Maybe she could try yoga, writing (stories or a journal), or calling a best friend on the phone (just set some limits).
Above all, remember that this is a tough time for girls at school too. Fourth and fifth grade is THE pressure grinder for kids. Too much is being asked of them for their stage of development, all while being told it will be worse in middle school (it isn't). Plus, their friendships are changing. Old friends are pairing off with others to become best buds, etc.
Try to find time to talk about what went on at school. And try to refrain from judging or giving too much advice -- especially from siding with friends or teacher. Even if she sounds unreasonable. You could just try a bit of "Wow... that sounds really tough. It sounds like that was really frustrating." If she asks why you think someone said or did something, then share. But otherwise, she just needs to feel someone is really listening to her and accepting her.
Hope this helps! BTW it sounds like you are doing a great job! Just "cheerfully" explain why you have to accomodate other people in the family as well, and tell her you wish you could [whatever she is demanding] but you can't. When she is not fussing up a storm, trying to give her a little special attention. Again, it's a lot like handling a tantrum prone toddler.
Make a special "date" with her to do something just the two of you.. Some part of her is probably grieving the fact that she feels so overwhelmed but doesn't have you as much as she'd like (as she did when she was very small). It can't be helped much, but having special times with you will ease the sting. And it will reassure her somewhat that she is still special to you.