Well, firstly, I must say I am not a teacher of English but just another language appreciator. Besides, I could not resist to try and help out my fellow Russian. I've been living in the US for over 8 years so, hopefully, my language intuition evolved enough to be somewhat useful.
1. While the second version, "People are very friendly here" is surely more grammatically accurate, I, personally, don't see why we can't say it the way your daughter did. Everybody would understand; the word order doesn't really seem as strict as we, in Russian schools, were taught; even in literature, not just in the spoken English... I wouldn't count it a mistake, anyway.
2. I, too, think it is incorrect. It is the same as to say "I want home" instead of "I want to go home." For us, Russians, it makes persfect sense to say "I'll be back after tomorrow" or "I want home" but in Englsih, I believe, it sounds like you miss the element that carries the main meaning.
Anyway, I would count it as a mistake, yes.
3. I wouldn't agree with the teacher on this one. "Truly yours", "Sincerely" both sound formal; yet for a signature in an informal letter, I see nothing wrong in signing like your daughter did. I, myself, receive letters from my American friends signed, "Your loving, such and such." "Your confused,..." "Your tired..." So it depends to whom you write, I guess... I don't think it is a mistake.
Well, hope if I didn't help, then at least lifted yours and your Daria's spirits.