I agree with jdtm.
As a person with a long history in elementary school (teacher, principal), I can tell you that trying to teach a 3 year old to write is not only a waste of time, but probably causing some emotional harm. What you want to be doing is reading to her a lot. Maybe teaching her to count. Not writing the numbers down! Why would you even be trying to have her write? Hopefully, this is not being done in the kindergarten. There is no public school that would admit a child at this early age - so hopefully, this is just a different name for child care.
The best way to teach her to read is to read appropriate books to her. Find a story she really likes and read it to her again and again. After a while, she will be able to start picking out the words. Do a lot of counting things " how many apples,etc". No writing.
As far as lunch - find out what the friend is snacking on- and try giving that to your daughter. If it is not food that you feel is healthy for her, then find a way to make her snack more interesting.
Also talk to her teacher (?), and get her ideas of what you can do to help her.
Frankly, I think your daughter might be too immature for school. I also don't think three-year olds need to learn how to write and forcing them will only backfire. As for eating the other child's snack, I am wondering if your daughter has not yet learned "boundaries and/or limits". But, back to my first thought ... I think she is just too young for school. (and personally, I think most children go to school far, far too soon). Anyway, just my "two cents"....