How should he deal with his apostate father and his mother who does not pray?

My father and mother have been married for 20 years. In the year 2001 my father went to Agadir in Marocko for some type of affair concerning some type of a restaurant, I cant recall the exact affair because I was 12 of age at that moment. Anyways, my father commited zina and my mother caught him with this sin which he commited on his return home, she did not divorce him due to my young age and my younger brother and sister, who were at that time 9 years old and 5 years old, we needed a father so she did this for her children, but she refused to sleep with him for a full year due to the pain she was suffering.
Please note that neither my father or mother prays, but my mother is on her way to perform salah inshAllah due to the muslim friends she has, as for my father, when he drinks alkohol, he used to speak badly about Allah and Muhammed s.a.w. in the most horrific of terms which I refuse to write here. The issue is sheikh, now that I am 20 years of age, my father again commited zina in 2008, I was 19 then and did not make salah. After that calamity I came to islam hamdiuAllah and sought His help and aid. My mother kicked him out this time but now, a year later he is back in our home. My mother uses an excuse like me now being engaged and soon getting married to bring him back, because she cant spend her life alone without a husband, because eventually my sister and brother will also get married, and she refuses to marry any other good man out there, so she chooses this garbage which I call him.
Me, now being muslim and knowing the punishment for this sin when one is married has motivated me in doing some actions. I have spitted in the face of the woman my father had sex with, she is russian, I called her by the worst of names in front of all the people, I have almost engaged in a physical fight with my father due to his action, I have threatened him and his lover, and as for my mother, I have risen my voice sometimes when I speak to her on this subject due to her strange choice of bringing this man back to her life after he has done this act once in 2001, and she forgave him, and now again in 2008, I think my mother is desperate, and I want my father out of the house, I have even considered kicking all his belonings out from our door.
Is my actions haram, how can I handle this situation, and how should I treat my mother and how should I act towards my father, should I treat him with respect after this act of his which he has not yet repented from because first of all he is a kaffir who does not believe in Muhammed.

Praise be to Allaah.

Firstly: 

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 meaning):

“...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”

[al-Mumtahanah 60:10]. 

There is no doubt that what your father is doing of reviling the Prophet (blessings and peace of Allah be upon him) is kufr and blatant apostasy from the religion of Islam. 

Please see the answer to question number 103082. 

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. 

Secondly: 

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, 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 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 141680. 

Thirdly: 

The fact that your mother does not pray poses a great danger to her religious commitment. Not praying because of laziness is major kufr according to many scholars. This opinion has been discussed in the answer to question no. 10094 and 5208. 

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, Ever Responsive. 

And Allah knows best.