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. We ask Allaah to exalt his mention as well as that of his family and all his companions.
First of all, we congratulate you for being guided by Allaah Who showed you the truth, and we ask Him to bless you with sincere repentance and accept it from you. There is no doubt that you are upon a great blessing which requires one to be grateful and thankful to Allaah, perform more acts of obedience and be keen on acquiring beneficial knowledge, attending circles of knowledge and keeping company with righteous women; these things will help you remain steadfast on the right path. Furthermore, you should know that Allaah's Mercy is vast and He forgives all sins [except Shirk –associating partners to Allaah]; Allaah Says (what means): {But indeed, I am the Perpetual Forgiver of whoever repents and believes and does righteousness and then continues in guidance.}[Quran 20:82]. Allaah further Says (what means): {Indeed, your Lord is vast in forgiveness.}[Quran 53:32]. Allaah even accepts the repentance of the non-Muslims, hypocrites and polytheists, so how is it that He will not accept the repentance of a Muslim who was lured by the devil who induced him to commit sins. Allaah Says (what means): {Say to those who have disbelieved [that] if they cease, what has previously occurred will be forgiven for them.}[Quran 8:38]. Allaah Says about the hypocrites (what means): {Indeed, the hypocrites will be in the lowest depths of the Fire, and never will you find for them a helper. Except for those who repent, correct themselves, hold fast to Allaah and are sincere in their religion for Allaah, for those will be with the believers.}[Quran 4:145]. Allaah also Says about the Christians who say that Allaah is the Messiah the son of Mary and say that Allaah is one of the three (what means): {So will they not repent to Allaah and seek His forgiveness? And Allaah is Oft-Forgiving, Most Merciful.}[Quran 5:74]. Therefore no one should despair from Allaah's Mercy, as despairing is one of the great major sins. Allaah Says about Ibraaheem (what means): {Ibraaheem (Abraham) said: "And who despairs of the Mercy of his Lord except those who are astray?"}[Quran 15:56]. So, despairing from Allaah's Mercy makes a person sinful. Thus, take the initiative to repent, and think good about your Lord. Allaah Says (what means): {Say: 'O My slaves who have transgressed against themselves (by committing evil deeds and sins! Despair not of the Mercy of Allaah; verily, Allaah forgives all sins. Truly, He is Oft-Forgiving, Most Merciful'.}[Quran 39:53]. For more benefit about the conditions of repentance please refer to Fatwa 87903.
As regards your father, if he is as you mentioned that he is anti-religion and that he deems that adultery and fornication are permissible, then he is in a very serious matter. So, he should be advised and reminded of Allaah with soft words and in a good manner. There is no harm to seek the help of pious and righteous people in this regard, and the help of those who could influence him. You should know that being kind, dutiful and obedient to him (in what is permissible) is one of the means of influencing him. You should not forget that all good lies in supplicating Allaah and seeking His Help, as He turns our hearts as He wishes. After all this, if he repents and returns to Allaah, all perfect praise be to Allaah, but if he persists as he was, then it is not permissible for your mother to stay with him, as he has indeed committed a sin by which he becomes an apostate. Your mother should take the matter to an institution which looks into the issues of Muslims, and you should be vigilant about him but still treat him kindly.
Finally, it should be noted that residing in a non-Muslim country leads to much afflictions and trials, so it is not permissible for a Muslim to reside there except for a necessity or a dire need. In this regard please refer to Fataawa 86405 and Fatwa 81464.
Allaah Knows best.