I do not understand what you mean by "medium school". I assume you want your daughter to go to a public elementary school in Stuttgart? In that case, just go to the nearest Grundschule... that's just a normal elementary school from Grade 1-4. Then, in the upper schools are Grades 5-13.
8am-6pm would be difficult... the school days here are pretty short (I'm a high school student, and the school days are shorter Germany than in America). I don't know how day care works around here, but there are plenty of young kids who get around alone here. I do not know where you are coming from, but when I lived in America, young kids never did stuff without parents. Here, I see plenty of young kids (under 10 years old) riding around the public buses, in the city alone. I can't offer you any advice here; this is just my observation.
School year here begins in Sept and ends in July.
I am not sure what exactually you are looking for. If you do not want to go to an international school, then you just go to the public school in your neighborhood. Very few kids go to private school in Germany. Actually, there are not many private schools at all. Almost everyone goes to a public school. All the public schools are similar in quality and actually, quite good.
The closest Grundschule to the Stadtzentrum is Jakobschule. (I'm not making a personal recommendation. Rather, that is the public Grundschule located closest to the Stuttgart Stadtmitte.) Here is a directory of Stuttgart Grundschulen: http://www.stuttgart.de/sde/menu/frame/top...bl/gen/2300.htm
. Under the listing for each school, it says "Schulbezirk". There you can see the region that the school serves.
Kindergarten... all I know is that kids go there normally between age 3-6, are often run by the city, but they are not free. On the other hand, offentliche Grundschule is free.
How old is your daughter? If she has already attended an "English medium school", then I think she would be too old for kindergarten?