Being a descendant of a person or a family depends on the definition of descendant. In general you are a descendant of all your grandfathers and grand mothers. But if you ask about being a Sayyed, then the definition of a Sayyed is the person who was born from a Sayyed father. So, your grandfather is related to the Prophet's family from his mother, but since his father was not a Sayyed, he is not a Sayyed. This is the verdict of most of our great scholars.
It isn't wajib for one's parents to live with them. Yes, one can live with the in-laws however in many cases it has caused issues in marriages. It is your right upon your husband to have a comfortable home and your own privacy. Yes, he can perhaps get his mother to move closer if he wants to have a good relationship with her but it wouldn't be recommend for her to move in if this would cause you discomfort. This should be voiced to your husband that it isn't obliged for you to accept this if it may cause issues.
It is unacceptable for a mother-in-law to be verbally abusive to her daughter-in-law. Verbal abuse, jealousy and hatred to that extent can be part of a personality disorder. I can recommend researching in detail the characteristics of the malignant, narcissistic woman and mother. In insulting someone you love, she is also abusing you. Mothers with a narcissistic personality disorder (as opposed to just being self-centred) are competitive in terms of who their child loves most.
How your children see you behave with your mother and wife will affect their own marriages down the line. Can you ask your mother not to verbally abuse your wife? If you feel you can't, that is revealing something about how your mother has trained you to relate to her, i.e. to remain passive and take the abuse; to not have enough self-worth to even politely ask her not to be verbally abusive.
Your duty is to love and protect your wife. You are the head of your household. You also have to protect the well being of your children. If they see their mother being abused their well being will be affected too.
According to Ayatollah Dastghayb-Shirazi, you are entitled to minimise or even cut ties with family members whose bad behaviour you can't reform, or whose bad behaviour gets worse by your presence, or whose bad behaviour you indirectly condone by co-operating with them. Being good to your parents does not mean condoning behaviour that could destroy your family.
You may want to look on youtube at videos on narcissistic personality disorder. Rages can be a symptom of that. There are coaches on-line that teach you how to protect yourself. I can recommend Dr Ramani, Anna Runkle, and Dr Daniel Fox. Look up the 'grey rock' method. Also be careful that you are not being used emotionally by your mother as a 'flying monkey', or that you are being 'parentified' (i.e. your mother is the child and you are the parent that feels you have to protect her). Not saying that this is happening, but it is something to be aware of.
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