From an audio exercise, both should be accepted. I have added it to the list of homophones but it may take staff a while to act on it.

"Arrivé" and "arrivée" are homophones. When a new sentence is added, a separate action is required to list homophones for the audio exercise. I have added it now but it may take staff a while to act on it.

When new sentences are added, any relevant homophones must be listed separately and then staff will take action so the variations are accepted on audio exercises. It is unfortunate but it is part of the growing pains when Duolingo expands the material in the course.

Because that sounds like the woman didn't begin to search until the moment you arrived, and it also doesn't convey the fact that her searching was not a one-time act, but had been taking place over an unspecified length of time in the past, which is the reason for the imperfective (l'imparfait in French): « Elle cherchait » au lieu de « elle a cherché ».

Reference to an action in progress (when interrupted by another action) is a classic use the past imperfect tense. "She looked for a book when I arrived" is not correct, but "she was looking...."

Yes, it should be accepted for a "type what you hear" exercise but there is often considerable lag time from making the request and staff taking action on it. It is in the works.

Because it is part of the compound past, which is how French speakers convey the perfective past "I arrived". Most verbs take the auxiliary avoir in compound past, but arriver, being a verb of motion, takes the auxiliary être. So, from « être arrivé » "to have arrived", it's conjugated as « (quand) je suis arrivé(e) ».

Of course, verbs have many forms (look at this table for all the ways you can conjugate arriver: https://en.wiktionary.org/wiki/arriver#Conjugation). But in this sentence? No, the passé composé is used because "I arrived" is a one-time act in the past.