How did concubines negotiate contact with non-eunuch men in & around the harem? I assume princes could interact with their moms, sisters, and other female blood relatives. Yet Mehmed III impregnating a servant while still a prince & Kosem caring for Osman suggests they weren’t 100% isolated from non-sultan men. How was this divide between princes, their “stepmoms” & other non-blood related women in the harem managed?
Royal children and royal consorts belong to the same family even if there is not a blood relation between them. This is something we'll see later, maybe influenced by European practices, but we have princes call their fathers' consorts "first mother" "second mother" "third mother" according to the women's ranks. So, these people — consorts and children — certainly had a relationship, especially after the practice of the prince governorate lapses and princes live in the harem before being secluded in the kafes.
Before the reign of Süleyman I, princes as young as 10/11 were sent to provinces away from Istanbul to learn how to rule. It's with him that for the first time actual adult princes live in the harem: Mustafa left Istanbul for Manisa at the age of 18, Mehmed was even older— 21 years old. In any case, princes weren't barred from talking to their fathers' consorts. There's actually a letter of Hürrem's in which she asks her husband to send her hello to Mustafa:
“If you send greetings to Sultan Mustafa, send him my note too.” Apparently she had included a separate letter to Suleyman’s eldest son in the scroll cylinder that carried her own to the sultan. The future would cloud Roxelana’s relations with Mustafa and his mother Mahidevran, but in 1526 there appeared to be harmony, or at least an effort on Roxelana’s part to keep up communication. — Peirce, Empress of the East: How a European Slave Girl Became Queen of the Ottoman Empire
Specifically to the examples you've mentioned:
the slave Mehmed III impregnated was part of Nurbanu's retinue, which means that he probably met her whenever he visited his grandmother (which royal grandchildren routinely did— daily in the XIX century, supposedly it was the same in the XVI century)
Osman was Kösem's step-son. Seeing as Mehmed was only a couple of months younger and they had therefore the same tutor, they could not not have a relationship of some sort. I also imagine that the children used to play together when they were little and the only sons of Ahmed I. The Venetian ambassador's remark that she was not to converse with him or his younger brother anymore does not mean that they wouldn't see each other anymore but that they weren't supposed to spend a lot of time together. Or alone. As Osman was motherless, he was "defenceless" because he didn't have anyone to look after his own interests; if Mahfiruze had been alive at the time, Kösem wouldn't have certainly taken him out for carriage rides with her children. They would have still seen each other on a daily basis, maybe for lessons or because Kösem would visit Mahfiruze (for tea, for example) but it wouldn’t have been that dangerous (from Ahmed I’s point of view, at least) because Kösem wouldn’t have tried to lobby the young prince
Hell, I'll even include Ahmed and Kösem who, according to Valier, met each other when the prince used to visit his own mother.
These things could happen, though it wasn't certainly a habit and it was forbidden to entertain an intimate relationship with a woman belonging to the sultan's harem, it doesn't matter her rank.
In the case of step-mothers and step-children though it was perfectly normal for them to have some sort of relationship. I mean, motherless children were entrusted to other consorts by the sultans themselves. In the XIX century, princes would routinely visit their fathers’ consorts on religious festivities and consorts with no children would even have favourites among the royal children: Mahmud II’s consort Tiryâl Hanımefendi was very close to the then-prince Abdülaziz, so much so that when he became sultan, she was considered a sort of second valide sultan. It’s true we’re much later in Ottoman history but in this case, Abdülaziz had a living mother.
I think the family dynamics of the Ottoman dynasty are very interesting.
