L.R.
Perry, this may sound a bit tough but I'm leaving it as is. The gist here is that your girls sound very smart, which is great, but very much on the road to total burnout. I respect your academically pushing them (we do the same to an extent) but hope you can back off before it's too late.
As others note - your girls are frankly spoiled, in terms of getting whatever they want. Please stop buying them what they want. This is very likely a strong reason your 10-year-old doesn't have friends; she is probably known at school as someone who is spoiled and possibly brags about what she does have. She may dismiss her possessions in front of you, but she clearly wants to appear like a rich kid around classmates: The fact she would ask for -- demand? -- a new suitcase for a school trip only because the ones you own aren't fashionable enough is a sure sign that she places a high value on the material things that other kids see she possesses.
But I see another huge problem here.
Your 10-year-old (and, assuming her schedule is the same, your 7-year-old) is vastly, utterly overscheduled and under tremendous pressure from incredibly high expectations on your part. Are your expectations really appropriate for her age or stage in life? Do you know what is normal for a 10-year-old, for instance, in terms of attention span, lack of attention to directions, etc.?
Your post makes me think you expect perfection. But when she does normal but annoying 10-year-old things -- not paying full attention to directions on a test, not listening the first time -- you sound as if you rapidly become furious and disappointed and let her know it.
She is being set up for some teenage years of huge rebellion if you don't let her be 10 now.
She attends full-day school but comes home and studies 3:30-7:00. That's an extra three and a half hours of study each day on top of school. From what you write, it sounds as if this study is not necessarily about doing her homework from school but about extra studying for outside exams you have her taking. You also talk about your frustration with her "carelessness" in these outside exams and even refer, your words here, to how "the end result comes out ***ty." Do you see how your words indicate you have astronomical expectations and how you are angry when she isn't perfect every time?
Please, please try tor re-read your own post and think like an outsider. Do you see how your daughter may be feeling huge pressure from you to be academically perfect? You are expecting her, basically, to do two schools a day-- the regular one and your own. You are expecting her to be "switched on" and in perfect study and test mode all day and up to one hour before her bedtime every night. When does she have "down time"? You want her to be socially popular and have friends, yet she has zero time -- not one hour -- to do anything other go to school, come home from school, start studying until 7:00, go to bed at 8:00. That leaves no time at all for her to slow down, have her own space, her own hobbies or thoughts, to have another child over or do any activity that is not part of her schedule.
You place huge emphasis on her being well-liked by teachers and on teachers relying on her. Again: She is the perfect student and perfect child. She is well aware that teachers like her -- and that they expect her to be a little teacher's aide. More pressure from them, even if it's well meant.
You actually have kids of 10 and 7 using a treadmill for exercise at home. If they had even an hour in the day that was free, they would not need a treadmill; they could move around and play.
She is on a treadmill of another kind -- a treadmill of extremely high, unattainable expectations. Your expectations. I have a smart kid too; we do extra studying for her tests; she goes to an advanced math program in the summers; we do extras like going to lots of museums and doing our own "units" of fun study in summer. She is in a science competition program that requires many extra hours a week outside school, for much of the school year. We expect a great deal of her academically and she knows it, and expects it of herself. But she also has outside activities (Girl Scouts, dance) and must have some down time each day to recharge. You can have high academic expectations without burning a child out, or filling a child's minimal free time with nice possessions.
Do your children have any activities that are not related to school or to studying for these outside tests? It's good that your child is in theatre and not solely in academics at school, but does she have anything outside school that she does that is just because she, not you, is interested in it? That she does because it is for fun or for exercise and not because it has some perceived higher value or will advance her academically? Does she have any hobbies, like to draw (not graphs or charts for school, but for pleasure), like to move around (has she done martial arts, dance, whatever?)? These things build the child's independence and also would help her find some friends with similar interests. But from your post, I would be concerned that if she took up dance you'd expect her to be the very best dancer very quickly, or if she did an art class, you'd expect her to be Picasso within a week. Maybe that sounds like I'm being hard on you -- but again, can you reread your post and see how people would get that impression?
You say you feel sorry for her "because she doesn't have friends and never goes out" but when is there time for it? Do you ever invite other children over? You say you're aware that both girls have too few chores and too many things, but do you ever tell the maid not to do chore X and then assign it to one of the girls, permanently, with no stuff as a reward?
Please consider talking to a family counselor about why you have always found it OK to just hand your children things; why you are giving them this level of academic work; and why your expectations are this high. And definitely please ask a family counselor about why you become so very, very angry at children these ages when they are not absolutely perfect.
Concrete suggestions: I'd suggest dropping the extra studying for a time, while increasing some weekly chores and replacing material objects with family outings -- do things together, rather than give the girls things. Tell them you are going to do a sweep of their possessions -- they will help -- and find things to give to charity. (That may be tough and they suddenly will just adore things they have not touched in months, but don't make it like punishment for them -- tell them it's because you want the WHOLE family to think more of others). Find them some kind of work to do for a charity; even a lemonade stand with all proceeds to a charity would be fine. And find them each one activity that has nothing to do with school and at which you (in your mind, not out loud to them) will say, "I am not going to say one word about how well or how badly my child does at this, but will let her enjoy it and/or fail at it on her own terms."