I am so sorry to hear of your mother’s death and pray that she will rest in peace.
In a Catholic Mass, music and Scripture readings ordinarily are considered ministries, and so are generally supposed to be filled by Catholics. That is why the lector (the one who reads from Scripture) and the cantor (the one who leads the singing) are supposed to be Catholics.
That said, it does seem like some accommodation could be possible to allow you to have some part in your mom’s funeral. While the Scripture readings must be read by a Catholic, your sister could solicit your (name removed by moderator)ut in choosing from among the selections of Scripture that the Church uses at funerals. While your non-Catholic friend should not be singing during the Mass, she might be invited to sing just before the entrance hymn. (Keep in mind that the song she sings should be approved first by the priest celebrating the Mass.)
I’m guessing that your sister is worried about making certain that the funeral Mass for your mom is “perfect.” She is also doing this while weighed down by grief. So perhaps she is not being as thoughtful as she might otherwise be. Perhaps you could suggest that you and she meet with the priest who will be presiding to discuss with him the possibilities for including you more in the arrangements. God bless.