Lineage of a child born through marrying one's mahram

Q: If a person married his mahram and children were then born from this relationship, will the nasab (lineage) of the children be established from the father? Likewise, will the ruling be the same in the situation where a person married a woman who was haraam upon him because of hurmat-e-musaaharah, ridaa'at, etc. Also, can mufti saheb please explain if there is any difference in the ruling if the husband knew that she was haraam or didn't know?

Proposing for a girl who does not speak to her father

Q: My question is that I met this girl and she and I have good intentions but I got to find out from her that her father treated her mother very badly and abandoned her and her children. So now she now she said that shew doesn’t consider her father a part of her life and when she meets him she doesn’t even talk to him.

1. Is she right to do this to her father? 

2. Can I ask for her hand in marriage from another person when her father is still alive just because she is not on good terms with him?

Hadith regarding preventing toddlers from attending the musjid

Q: Does the following Hadeeth have any weaknesses attached to it?

Wathila Ibn Al-Asqa' (Radiyallahu Anhu) narrates that the Messenger (sallallaahu alaihi wa sallam) said, "Keep away from your masjids your children, your lunatics, your buying, your selling, your disputes, raising your voices, implementation of your hudood and withdrawing of your swords. Make provision at its doors for fulfilling needs of purity; and perfume it with incense on Fridays." (Sunan Ibn Maaja, hadith 750)

If so, can it still be used for advising/preventing toddlers from coming to the masjid?