What is the ruling on adult male children from a divorced wife living with the new wife in the same house, and what is the ruling on khalwa (private seclusion) in this situation?

The husband's sons are considered Mahrams to his wife, and it is permissible for them to be in seclusion together, and she is not required to observe hijab in front of them.

Verse (31) of Surah An-Nur mentions the permissibility of displaying adornment to the sons of husbands, because they are in the position of sons in terms of Mahramiyyah.

And anyone whose marriage is permanently forbidden to her by lineage or breastfeeding is considered a Mahram.