Translation:She was looking for a book when I arrived.
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 ».
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.