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.
Adultery is one of the greatest and most obnoxious sins. Generally, a person is dragged to it by being too tolerant about the causes that lead to it, like being in seclusion with a woman, forbidden mixing and so forth. One of the most dangerous of these causes is failing to observe the religious requirements when employing a female servant at home. So you have to sincerely repent to Allaah, perform good deeds as much as possible and avoid any reason that could lead you to commit adultery again.
As regards the other issues that you raised in your question, the answer will be as follows:
1) The jurists differed in opinion with regard to marrying a pregnant adulteress. The Maaliki and Hanbali schools are of the view that it is not permissible to marry her [the marriage to her is invalid]. However, the Hanafi and Shaafi'ee schools are of the view that it is permissible to marry her. So according to the second opinion, the marriage that is mentioned is valid but it is not permissible for this man to have sexual intercourse with her until she gives birth to the baby. The evidence is the narration of the Prophet as he said: "It is not permissible to have sexual intercourse with a pregnant woman [when getting married to her] until she gives birth to her child." [Abu Daawood]
2) When this woman gives birth to her child, the child is considered as being born out of an illegal sexual relationship so he has to be named after his mother. It is not permissible for you to name him after you as adoption is forbidden in Islam.
3) You have to conceal the sin of this woman, so you should not inform anyone that she committed fornication or that she is pregnant, except her husband. It is an obligation to inform the latter, because there is some religious harm in concealing her sin from him, which is that he might have sexual intercourse with her while she is pregnant and this is forbidden; in addition to this, it could be that he would name the child after him but he is not the father.
4) It is not permissible for you to marry her since she is married, but if her husband divorces her or dies, then it is permissible for you to get married to her after the end of the waiting period provided she has repented from her sin.
Allaah knows best.