All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
If the deceased did not leave any other heir except those mentioned in the question, then his wife gets one-eighth of the inheritance as her legal share due to the existence of direct heirs (children); Allaah says (what means): {But if you leave a child, then for them [i.e. the wives] is an eighth of what you leave, after any bequest you [may have] made or debt.} [Quran 4:12]
The remainder will be divided amongst the sons and daughters by Ta’seeb (by virtue of having a paternal relation with the deceased and not having an allotted share, so they get what is left after the allotted shares have been distributed); the male twice the share of the female as Allaah says (what means): {Allaah instructs you concerning your children [i.e. their portions of inheritance]: for the male, what is equal to the share of two females.}[Quran 4:11]
Therefore, the inheritance should be divided into 120 shares, the wife gets one-eighth, which is 15 shares, each son gets 14 shares, and each daughter gets 7 shares.
As regards the house whom the deceased had written in the name of his wife, then if he had done so as a will that she takes after his death, then this is a will for an heir and it is forbidden in Islam. This will cannot be effective except after the consent of all the other heirs for the Hadeeth of Abu Umaamah Al-Baahili who said: “I heard the Messenger of Allaah saying: 'Verily Allaah has given every rightful person his right, so there is no bequest for an heir.' This was related by Ahmad, Abu Daawood, Ibn Maajah, and At-Tirmithi. Ahmad and At-Tirmithi deemed it Hasan (good). Ibn Khuzaymah and Ibn Al-Jaarood deemed it strong. Ad-Daaraqutni reported it from Ibn ‘Abbaas but added at the end of it: 'Unless the heirs agree to this'. Ibn Hajar considered its Isnaad (chain of narrators) to be Hasan.
Therefore, if your half-brothers from your father's former wife do not agree to make this will effective, then they have the right to take their share from the house and it is not permissible for your mother to prevent them from their right.
On the other hand, if the deceased had written the house as a Hibah (gift) to his wife, not as a will to be effective after his death, then if he had gifted it to her during his death sickness, then this gift is not effective because the gift during death sickness takes the same ruling of the will [i.e. not effective and forbidden]. Ibn Al-Munthir said: “All the scholars whom I know their statements agreed that the ruling of gifts during the illness in which the donor died has the same ruling as the will; this is the view of Al-Madeeni, Ash-Shaafi’i, and Al-Koofi.” Since it has the same ruling as a will, then the previous explanation applies to it that it is not effective except with the consent of the other heirs.
However, if the deceased had gifted the house to his wife while he was in good health and he was not during an illness that might cause him to die, then a man gifting the house where he lives to his wife is not considered an effective gift until he himself goes out of it (the house), and takes all his belongings from it and she takes possession of the house and disposes of it as if it was her own property, so if he had not done so and he lived in it with her until he died, then it becomes inheritance that should be divided among the heirs like all his other wealth, and in this case the gift is considered void as stated by some jurists.
Allaah Knows best.