Is it allowed for a divorced man with a 10-year old son who's living permanently with him to marry a divorced woman with a 3-year old daughter who also lives permanently with her? Can both kids be raised together in one house after the marriage?
Marriage between a man and woman is inherently permissible, unless:
- They are related within the prohibited degree (mahrams).(1)
- The woman is already married (not divorced). (2)
- The woman is a non-muslim and neither Christian nor Jew, or the man is a non-muslim.(3)
- The man and woman have been previously married and thrice divorced (they will not be allowed to remarry unless the woman marries and is genuinely divorced by another man).(4)
Previous kids are not a cause that prevents marriage. The Prophet(SAW) married Khadijah(AS) while she had children from previous marriages, and he married other wives while he had children from deceased Khadijah.
It is not a recommended practice to keep children with step-parents since they are usually not loved the same way by their step-parents as their blood-relations.
The custody of an underage child belongs by default to the mother when she is a divorcee or widow, though if she remarries the right of custody goes to the maternal grandmother or other female relatives who are not married to 'stranger' for the child. An older child has the right to choose their custodian. (7,8)
As long as the daughter hasn't reached puberty, there is no need for any separation from her step-brother(9,10). After puberty however, the children will not be mahrams to each other, and must cover themselves as they cover themselves from a stranger ... though the children will be mahrams to the step-parents(11).
The children have nothing to do with it. It only complicates things. Because the children don't become Mahram.
Uma's answer is very good, I just want to give you an interesting touchstone.
Zayd ib Harithah was prophet Muhammad's step son.
مَّا كَانَ مُحَمَّدٌ أَبَا أَحَدٍ مِّن رِّجَالِكُمْ وَلَٰكِن رَّسُولَ اللَّهِ وَخَاتَمَ النَّبِيِّينَ ۗ وَكَانَ اللَّهُ بِكُلِّ شَيْءٍ عَلِيمًا
Muhammad is not the father of any of your men, but (he is) the Messenger of Allah, and the Seal of the Prophets: and Allah has full knowledge of all things.
The prophet wasn't his real father. In Islam, blood (+ marriage) ties are what bond people. Which is why adopted children don't inherit. (The deceased can give away as much he/she wants to anyone, but by default Islamic law they don't inherit anything, they are to inherit for their biological parents).
For this situation there is NO blood tie between the two children. So they aren't Mahram.
There is only a blood tie between the son and his step-mother through his father, so boy and step-mother are Mahram, and a relation between the daughter and her step-father through her mother, so daughter and step-father are Mahram.
If a new child is born, then that new child is Mahram to everyone :D.
(Kinship is an answer to many modern questions: inheritance, homosexuality, adoption, guardianship, Mahram vs. non-Mahram, friendship, etc.)