I have some schools that ask questions about myself, like “Explain a time your perspective was challenged” and etc. Should I connect back to the school in some way for these essays, or just focus on my own story like my personal statement.
Does the prompt ask you to connect it to the school? There’s your answer.
No, you do not need to bring each essay back to how X University will be a great place to explore this further or will be the light at the end of the tunnel for a situation you are dealing with…
They genuinely want to get to know you a little better. They actually read them and it would be boring if every essay just talked about what a great place they work. And remember they are not trick questions.
They do not want contrived answers, but if there really is a Ted talk you saw with a professor from X college that inspired you to think differently about a situation you faced, then maybe that is ok.