The first two sentences mean almost the same thing, but not quite.
Sentence 1 implies that she has just joined, or is in the process of joining, the school. The speaker is asking about the girl's long-term plans.
Sentence 2 implies that the speaker has heard, or thinks it possible that, the girl is soon to leave the school. The speaker is asking about medium or short-term plans.
The future perfect normally talks about an action that will be complete in the future. "By this time tomorrow, I will have finished the project." This means that this time tomorrow, you will be able to say, "I have finished the project."
The question "Will she have stayed in our school by the end of the year?" is a bit unusual, but possible. It means this: "At the end of the year, we will be able to say: 'She has stayed in our school' -- is this statement true?"