Brother-in-law has a mistress

Dear PAO,

I want to ask your office about my sister’s husband who had a relationship with another woman while already married to my sister. To make it worse, it was rumored that my brother-in-law has a child with his mistress. My sister now wants to ask whether his mistress can obligate my brother-in-law to give financial support for their supposed child even if the said child doesn’t use my brother-in-law’s surname. Also, is it possible for the child of his mistress to have my brother-in-law’s surname in the child’s birth certificate? Will it be an issue if my brother-in-law delays the acknowledgment of the child until it can be confirmed that the latter is really his? Is it also possible for my sister to file a case against my brother-in-law’s mistress for their affair and disruption of my sister’s family life?