Praise be to Allaah.
You have to keep calling your sister to Islam, because Allaah says (interpretation of the meaning):
Invite (mankind, O Muhammad) to the way of your Lord (i.e. Islam) with wisdom (i.e. with the Divine Revelation and the Quraan) and fair preaching [al-Nahl 16:125].
You have to bear your mothers insults with patience, as Allaah commanded His Prophet Muhammad (peace and blessings of Allaah be upon him) to do, when He said (interpretation of the meaning):
So bear patiently (O Muhammad) what they say, and glorify the praises of your Lord before the rising of the sun, and before its setting [Ta-Ha 20:130]
Do not feel sad because your sister has turned away, or regret the time you have spent in calling her to Islam if she does not respond. As Allaah said to His Prophet Muhammad (peace and blessings of Allaah be upon him) (interpretation of the meaning):
So destroy not yourself (O Muhammad) in sorrow for them) [Fatir 35:8]
Perhaps you would kill yourself (o Muhammad) in grief, over their footsteps (for their turning away from you), because they believe not in this narrated (the Quraan). [al-Kahf 18:6].
As for your answers to her questions, they are correct. With regard to waking up at night, although Allaah takes peoples souls when they sleep, there is no contradiction between the two, for the One Who is able to take the soul is able to put it back. This is why it is mustahabb for the person who wakes up from sleep to say: Al-hamdu Lillaah illadhi radda alayya roohi wa afaani fi jasadi wa adhina li bi dhikrihi (Praise be to Allaah Who has returned my soul to me, kept my body healthy, and given me permission to remember Him).(Classed as Hasan by al-Albaani, Saheeh al-Jaami, 1/329)
With regard to the inheritance of women as you say, women are not responsible for spending on others; that is the duty of men, who have to pay the mahr and provide accommodation for their womenfolk, etc. Women wait for someone else to take the initiative, unlike men. So it is not fair for them to receive the same share in the case of inheritance. Allaah is the most just of those who are just and the most wise of those who pass judgement.
I ask Allaah to make you steadfast (in Islam) and to reward you for your dawah. You have to be gentle with them and not do anything to provoke them. Do not insult their religion, for that could provoke them to insult the Prophet (peace and blessings of Allaah be upon him). And Allaah is the source of strength.