Praise be to Allaah.
If your father is not Muslim, or he was a Muslim but he
apostatized from Islam -- which is what may be understood from your question
-- it is not permissible for your mother to go back to him at all, until he
announces his repentance and enters Islam once again.
Allah, may He be exalted, says (interpretation of the
“...if you ascertain that they are true believers send
them not back to the disbelievers. They are not lawful (wives) for the
disbelievers nor are the disbelievers lawful (husbands) for them”
There is no doubt that what your father is doing of reviling
the Prophet (blessings and peace of Allah be upon him) is kufr (disbelief) and blatant apostasy from the religion of Islam.
Please see the answer to question number
Hence it is not permissible for you to let him return to the
house and live there with your mother; throwing him out of the house is the
least to be done with someone like him.
Your father's apostasy from Islam means that all the rights
of honour and good treatment that he had over you no longer apply. The apostate has no respect or protection according to sharee‘ah (Islamic law), and he does
not have the rights of kinship and kindness. Rather he should be shunned and ignored, except for the sake of da‘wah (inviting to Islam)and offering advice.
However, it is not wise to get involved with arguments and
disputes with him, especially in front of people.
See the answer to question number
The fact that your mother does not pray poses a great danger to her religious commitment. Not praying because of laziness is major kufr (disbelief) according to many scholars. This opinion has been discussed in the answer to
question no. 10094 and
Hence you must strive hard to advise and exhort her, and to
remind her of the rights that Allah has over her, the obligatory nature of
prayer, its status in Islam and the seriousness of not praying.
Let your da‘wah to her be with beautiful preaching and
kindness, especially since she is on her way to becoming committed to
praying regularly as you said.
We ask Allah to guide your parents and accept their
repentance, and to help them to do all that is good, for He is All Hearing,
And Allah knows best.