Praise be to Allah.
The basic principle concerning the estate of the deceased is that it should be shared out among the heirs after the person has died, and after the funeral expenses and any outstanding debts to Allaah (e.g., zakaah) and other people have been paid, and any specific bequests that may have been made have been dealt with. It is disliked (makrooh) to delay sharing out the estate, because of the harm that this may cause to the heirs, but if all the heirs agree to delay it, there is nothing wrong with this. If your mother was a partner in your father’s business, then she should take her share of the company, then she is entitled to one-eighth of your father’s estate, then with regard to you and your siblings, the males are entitled to the share of two females, if the deceased’s parents are no longer alive. If you and your mother want to leave everything as it is, and let the business continue operating, there is nothing wrong with this; it is clear what each person’s share is, and whenever one of you wants to ask for it, he may take it.
It is not permissible for the mother to give something to one of her children and not the others, because of the hadeeth narrated by al-Nu’maan ibn Basheer (may Allaah be pleased with him), who said: “My father gave me some of his wealth as a gift, and my mother said, ‘I will not accept this until you ask the Messenger of Allaah (peace and blessings of Allaah be upon him) to be a witness to this.’ So my father went to the Prophet (peace and blessings of Allaah be upon him) to ask him to be a witness to the gift, and the Messenger of Allaah (peace and blessings of Allaah be upon him) said, ‘Have you done this for all your children?’ He said, ‘No.’ He said, ‘Fear Allaah and treat all your children fairly.’ So my father came back and took back the gift.” (Agreed upon).
For more detail on this topic and answers to the rest of your question, please see Question #1151. It is not permissible for a mother to bequeath anything to one of her heirs, because the Prophet (peace and blessings of Allaah be upon him) said: “There is no bequest to the heir.” (Kashf al-Qinaa’, 4/342; Ghaayat al-Muntahaa, 2/335; al-Mughni, 5/604). If she does, her bequest should not be carried out.