As salaamu alaykum,

Yeah, it's allowed to recite the Quraan anywhere Helena except for the toilet, we're not supposed to make Dhikr in there regards to the hadeeth that the Prophet sallaahu 3alayhi wasallam was in the toilet and a sahaabiy passed him by and gave him salaams , the Prophet sallaahu 3alayhi wasallam did not return it.. and when asked why he did not reply he said: Verily this place is not a place that Allah's name shall be made remembrance in (hadeeth; by meaning..)

As for reading it out loud Allahu a3lam but I think women should avoid doing so because when we read the Qur'aan we have to beautify our voices, and we cannot do so in the presence of our Non-Mahram..

Just my opinion Ukhtiy I ain't no scholar!!