Do illegitimate children inherit after their parents get married

Q: A Muslim lady got married to a non Muslim and had 3 children from him. If the marriage is not valid and the children are considered to be illegitimate and not inherit from their father even if later on he becomes a Muslim, are such children still considered to be illegitimate? Will they inherit equally with their brothers who were born after the fathers conversion?

Dream

Q: I saw in a dream that in the sky was written "laa illaaha illallahu mohammadur rasool Allah" and on the right side I saw Allah and on the left side Mohammad. The sky was very blue. What is the meaning of this? 

Women who were ungrateful to their husbands

Q: The hadith says the majority of people in hell are women who were ungrateful to their husband's. I'm divorced, I did everything for her, she cheated on me, stole from me, I always gave her dawwah but she ignored. I'm not saying I want her to be punished. I just want to be compensated in the next life. However does that hadith only mean a wife who continues to be married or both?

Borehole water mixed with the sewer line

Q: I have a small well in my home called borehole. Recently i've come to know that the water changed it's smell and this is due to maybe the outside line of guter/drain is mixing in the well water although the well is quite deep almost more than 40 feet. What would you advise as per the water? Has it become impure or can it be used for bath and other purposes?

Keeping contact with a married woman

Q: I am in love with one girl for the past 10 years. 5 years ago she got married to another person and she is not happy with her husband because he beats her daily and he is not keeping her happy. She still loves me and she is in contact with me. She says that she can't live without me. She wants to marry me but her husband is not giving her a divorce. She said that she will leave her home and she will marry me. Is this nikaah valid in Islam and can we marry and can be happy inshallah. What is the right way and procedure.

Dream

Q: One of my friends saw in his dream that one hazrat was standing and that hazrat called my friend and said that I will introduce you to someone who is the king of this world. He introduced him saying that this is prophet Mohammed (sallallahu alaihi wasallam) and my friend spoked to Rasoolul Allah sallallahu alaihi wasallam. And he said to my friend to follow my words and follow my sunnah... Can u tell me if this is good?