In all practicality,

 

your neighbor is asking a favor of you, not just your daughter.

There should be a kind of cat feeder that dispenses food at meal time; you only need to fill the feeder once in a few days.

Go with her if you are not comfortable that she gets into an empty house by herself.  Take a walk around the house before going in.

They give you the key. It means they trust you. Think highly of yourself.