Praise be to Allah.
First of all, you should hasten to enter Islam, which is
easy to do. An explanation of what is involved may be found under Question 114.
(See also Questions 122, 158
Secondly, if this lady becomes a Muslim or one of the People of the Book (Christian or Jewish), repents to Allaah for the forbidden relationship, and remains chaste, then it is permissible for you to marry her.
Thirdly, the children who were born to this woman outside of marriage are not your children; they cannot take your name and must take her name. However, this should not prevent you from treating them kindly, taking care of them and spending on them. In the future, you should try to call them and their mother to Islaam.
Fourthly, going to her church and attending rituals of kufr is not advisable, because of the possible negative consequences. We advise you not to do that, but this should not be any obstacle to your entering Islam. If you were to say that it is a choice of either embracing Islam and going to the church with her, or not embracing Islam at all, we would definitely tell you to embrace Islam. But we would like you to think of the meaning of these two soorahs of the Qur'aan (interpretation of the meanings):
"Say: He is Allaah, (the) One.
Allaah-us-Samad (The Self-Sufficient Master, Whom all creatures need; He neither eats nor drinks).
He begets not nor was He begotten;
And there is none co-equal or comparable unto Him."
"Say: O Kaafiroon (disbelievers in Allaah, in His Oneness, etc.)!
I worship not that which you worship,
Nor will you worship that which I worship.
And I shall not worship that which you are worshipping.
Nor will you worship that which I worship
To you be your religion, and to me my religion (Islaamic monotheism)."
Finally I advise you, a wise man who is striving to make the right choice, hasten to enter Islam and practise it: Allaah will help you to overcome all obstacles. May Allaah take care of you.