First of all, let me express my sincere condolences to you. I pray to Allah in the manner of the Prophet (peace be upon him) who used to console the bereaved in this way: "May Allah console you, inspire you with patience, and forgive your beloved who has passed away."
I know it is hard on you, but you should accept Allah's will and do what you can on behalf of your dad. Although du'a is the most important thing, you should do for your dad daily, you may also do other things on his behalf whereby he may be rewarded.
For further details on this, let me cite from one of my earlier answers on a similar question:
"Our duties towards our parents, as we learn from the sources, do not cease with their death; rather they continue as long as we live. In a report cited by Imam al-Bukhari in his famous work al-Adab al-Mufrad we read, "A man approached the Prophet asking, "Is there anything I must do in terms of kindness towards my parents after their death?" The Prophet replied, "Yes, there are four things for you to do: Praying and asking forgiveness of Allah on their behalf, fulfilling their promises, respecting their friends, and fostering their ties of kinship..."
Moreover, it is highly recommended for us to visit the graves of our parents. Such visits serve as excellent reminders for us besides prompting us to remember and pray for our parents. The Prophet (peace and blessings be upon him) said, "In the past, I had forbidden you from visiting graves, but now you should do so, for it might remind you of the next world."
After enjoining upon us the duty of excellence in rendering kindness towards our parents, Allah orders us in the Qur'an (Surat al-Isra', 17: 24) to pray for our parents saying, "Our Lord, have mercy on our parents, even as they had mercy on us, while we were little!"
Lastly, there are traditions from the Prophet (peace and blessings be upon him) that indicate that often a person in the grave suddenly finds his station elevated or conditions change, and when he enquires why it is happening to him, he is told that it is by virtue of the good deeds or prayers of his offspring on his behalf.
Therefore, in light of the above, I urge you not to be slack in praying for your mother besides doing whatever good deeds you possibly can on her behalf."
As for your mom, she should observe the 'iddah or waiting period of four months and ten days. However, during the 'iddah, she can move around-if needed. She may also live with her children, if she cannot live alone.
You may also consult the following answer.