"She exits through the atrium."
Translation:Per atrium exit.
Ea is anaphoric: it refers to somebody who was already mentioned. It can be translated by she, or the with a noun: ea femina "the woman (we are talking about)".
Haec, ista and illa have a demonstrative value, they refer to somebody in space:
— haec femina: "this woman (who is near me)";
— ista femina: "that woman (who is near you)";
— illa femina: "yonder woman :-) (who is far of us)".
To answer your first three questions, the Latin word exit (3rd person singular active indicative conjugation of the verb exeo, exīre) can mean any of the following in English, depending on context: "she exits", "she is exiting", "he exits", "he is exiting", "it exits", "it is exiting." The Latin phrase per atrium can mean any of the following: "through an atrium", "through the atrium."
This gives a minimum (I may have missed some obscure alternative translations) of 12 English translations for the Latin per atrium exit. One of these options is "she exits through the atrium."
However, ea per atrium exit also translates to "she exits through the atrium" so that should be an accepted answer if the exercise is to translate that phrase from English to Latin.
This is a major consideration when translating any language, not just Latin. Certain information can be conveyed differently by different languages and learners should be aware that some things will not be what they are used to. In this case, English speakers cannot simply use a conjugated verb because we must use a noun or pronoun or whatever plus a conjugated verb ("she exits" instead of just "exits"); Latin does not have this same requirement. Simply using exit without an explicit subject is not uncommon when reading classical Latin works. It can be confusing and frustrating but that's part of the language.