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, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
If the deceased did not leave other eligible heirs entitled to inherit other than his wife and two sons, then the deceased’s wife gets one eighth of the estate as a fixed share because the deceased has descendants entitled to inherit. Allaah, The Exalted, says (what means): {But if you leave a child, then for them (wives) is an eighth of what you leave, after any bequest you [may have] made or debt.} [Quran 4:12]
The remainder of the estate should be equally divided between the two sons by virtue of Ta‘seeb (i.e. 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 Prophet, sallallaahu ‘alayhi wa sallam, said: “Give the Faraa'idh (the shares of the inheritance that are prescribed in the Quran) to those who are entitled to receive them. Then whatever remains, should be given to the closest male relative.” [Al-Bukhari and Muslim]
Therefore, the estate should be divided into sixteen shares; the wife gets one-eighth of the estate (2 shares) and each son gets seven shares.
As for what you mentioned in the notes section in the question, then in principle, the car, house, money in the bank account, and all the properties are considered the property of whomever they are registered under his name. If the house and car are registered under the name of your father and his wife (joint ownership), this means that these properties are owned by them both. If you claimed that the wife does not have joint ownership and that they are the property of your father alone, then you are obliged to produce evidence supporting your claim, unless the wife endorsed it. Ibn ‘Abbaas reported that the Prophet, sallallaahu ‘alayhi wa sallam, said: “Were people to be given everything that they claimed, men would (unjustly) claim the wealth and lives of (other) people, but the oath must be taken by the defendant (in the absence of any proof against him).” [Al-Bukhari and Muslim]
Al-Bayhaqi narrated with a Saheeh (sound) chain of narrators that the Prophet, sallallaahu ‘alayhi wa sallam, said: “Evidence must be presented by the the claimant, and the taking of an oath is incumbent on the one who denies." As-San‘aani wrote, “The Hadeeth indicates that a person’s mere claim should not be simply accepted; it has to be supported by evidence or the defendant’s endorsement.” [Subul As-Salaam]
The accepted evidence in proving the ownership of property is having at least one man and two women as witnesses or a man and the oath of the claimant. Ibn Qudaamah wrote, “The testimony of a man and two women or a trustworthy man (as a witness) with the oath of the plaintiff are the minimum evidence accepted in cases involving property. . . . Most scholars believed that one witness and the oath of the plaintiff in property disputes are adequate. This has been reported about Abu Bakr, ‘Umar, ‘Uthmaan, and ‘Ali . This is the opinion held by the seven jurists (of Madeenah) ‘Umar ibn ‘Abd Al-‘Azeez, Al-Hasan, Shurayh, Iyyaas, ‘Abdullaah ibn ‘Utbah, Abu Salamah ibn ‘Abd Ar-Rahmaan, Yahya ibn Ya‘mur, Rabee‘ah, Maalik, Ibn Abi Layla, Abu Az-Zinaad, and Ash-Shaafi‘i .” [Summarized from Al-Mughni]
Based on that, if you produce evidence proving your claim, then the deceased’s wife has no right to more than her due share as a wife, namely, one-eighth of the estate. This means that she is entitled to only one-eighth of the house, car, and all the deceased’s properties as long as the deceased has descendants entitled to inherit. In this case, it is impermissible for her to take more than her due share by means of man-made law, because these are the rights of other heirs. Man-made laws do not deem unlawful what is lawful (on the basis of the Sharee‘ah) nor do they deem lawful what is unlawful (on the basis of the Sharee‘ah). When it comes to distributing the inheritance, Muslims should abide by the Sharee‘ah inheritance laws and not man-made ones.
If you fail to present evidence proving your claim and the wife does not endorse you, then she is entitled to whatever has been registered under her name from the deceased’s properties and they should not be included in the estate. In addition, she is entitled to one-eighth of the properties registered under the deceased’s name as well.
Allaah Knows best.