"She opened the door."
Translation:Ella abrió la puerta.
"Abierto" is a past participle. It indicates the already-"opened" state of a thing ("the door was open") not the act of opening ("she opened the door"). Often the same word in English, but not in Spanish.
You may be thinking of the indirect pronoun 'le'. In Spanish, it's common for both an indirect object pronoun and the noun to which it refers to appear together, which you rather did for the direct object 'la puerta'.
Additional concise information can be found here: http://www.spanishdict.com/guide/indirect-object-pronouns
Abría is like "was opening", it´s an action that takes some time, it´s different from "abrió" (she did it once).
Nuevamente depende del contexto ya que abría es copretérito y abrió es pasado simple.
No. Verb endings are determined by tense, number, and person (first-, second-, or third-person, with second having some variations for formal vs. informal speech). Gender doesn't enter into it.