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.
Your question included many points, among which are the following:
1- With regard to your question: “Is it Haraam (unlawful) for me to stay with my wife until I can find a better wife?”; the answer is that since she is considered a Muslim, then this is not unlawful for you as long as you have not divorced her and she has become irrevocably divorced from you. Besides, it is not permissible for you to desert her in a way that she is harmed. Therefore, you should not leave her stranded; if you do not keep her in kindness, then you should divorce her in kindness.
On the other hand, if she is disobedient, it is permissible for you to discipline her in an Islamic manner as clarified in Fatwa 85402.
Moreover, the scholars determined the period of deserting of one’s wife due to her disobedience as one month, and this should be deserting her in bed only [i.e. refusing to have sexual intercourse with her].
2- With regard to your question: “Do I have to keep her or divorce her?” the answer to this question is that this depends on the benefit. If you expect that she would become righteous or that you fear harm on her by her apostatizing from her religion, then, in this case, it would be better to keep her and be patient with her and endeavor to reform her. Otherwise, separating from her is better. Indeed, you may find another wife who would accept to marry you despite your health condition and the fact that you do not look attractive as you mentioned.
As regards that child, he/she is not attributed to you as he/she is born out of wedlock; rather, he/she should be attributed to his/her mother and it is the mother who is responsible for that child. For more benefit, please refer to Fatwa 85430.
3- With regard to your question: “Is she really a Muslim or not?” the answer is that if she embraced Islam, she is considered a Muslim unless she commits something that necessitates her apostatizing from Islam. If it is confirmed with certainty that someone is a Muslim, then it is not permissible to judge him to be as a non-Muslim except with certainty as well. For more benefit on the conditions that should be met for declaring someone as a non-Muslim, please refer to Fataawa 91567, 84943 and 135063. Therefore, the fact that she does not observe some obligations does not mean that she becomes a non-Muslim due to that.
Finally, it should be noted that loving a non-Muslim because of him/her being a relative and the like while hating the religion he is following, is permissible. However, the obedience of the wife to her husband takes priority over her obedience to her parents, and his right takes priority over their rights, let alone over other people’s rights.
Allaah Knows best.