Yes is sounds as if this behavior is 'normal', the fact she is okay at school tells me if she did have a problem it wouldnt simply surface at home, children do seem to be having emotions at an earlier age than we did, does she get plenty of positive attention and is her Dad involved ,with games and sports and outside activities, how about distraction from her 'woes' and focusing on things she enjoys ,dont feed into the negative behavior if she gets mad let her and if she doesnt get attention from it it may disappear.
She's very shy around others and no one would suspect how difficult she is at home when she feels comfortable.
The above sentence reminds me of our child. Our child suffers from anxiety (although it is much more severe than what you are describing). I am wondering if your daughter is suffering from social anxiety, which by the way, is often co-morbid with depression. Frustrations and tantrums in the safety of the home are very common behaviours of children suffering from anxiety.
I might suggest you google the words/phrases as "childhood anxiety" or "shyness in children" or "social anxiety in children" or similar in order to find out more about this issue. It might be wise to discuss this issue with your daughter's physician as anxiety often exacerbates during puberty (and your daughter is not far from that life event). Perhaps he/she might be able to offer suggestions on how your daughter might be better able to cope and manage her fears/anxieties. If your daughter's fears/anxieties increase, then you may wish to seek a referral from your family doctor to a medical specialist with experience in anxiety disorders. By the way, anxiety is a very common mental health issue and is highly treatable. I wish you the best ....