(Repost after my first post was removed for bad wording)
Sorry if this question is in poor taste but it's something that I can't get my head around. Historically, I can imagine that many bastards (especially ones born to prostitutes and the like) would have been completely ignored by their fathers, if they were even aware of their existence at all. But I can't help but imagine that bastards that are born in their father's own household may have been treated differently, as it's not so easy to just ignore them. It's a historical fact that most freeborn Romans who could afford to do so owned slaves, and as far as I am aware there were no legal or social restrictions in having sex with them (well, rape) and getting them pregnant. Surely this must have happened a lot.
I'm very curious about what life would have been like for a child born to a slave mother and slave-owner father, who might grow up in their father's home alongside their father's legitimate children, and are considered part of the household and familia, as is my understanding of how Roman slaves (at least, domestic slaves) were seen. Would their father have any particularly paternal feelings towards them, or would they be seen as just another slave/asset and have no relationship to their biological father outside of the slave/master one? Would they remain a slave, or might their father be inclined to free them, or at least treat them better than other slaves? Or are they actually more likely to be mistreated, sold, or exposed? Would they be aware that their master is their father? How are the wife and (legitimate) children of the family likely to view such a child? And the child's mother for that matter? Were there any social norms or laws regarding getting your slaves pregnant, or the treatment of your own children born into slavery?
Sexual abuse is a major presence in ancient slavery (and in most systems of slavery). To start with your question regarding Roman laws and attitudes toward sexual use of enslaved people, it was considered fully within slaveholders' rights to abuse his slaves and it was an accepted social norm. Laws that speak of this negatively are in reference to men who abuse another man's slave, which was considered an affront against the holder, not the enslaved person. The head of household was considered to have total power over the enslaved bodies in his household--even if that meant physical or sexual abuse. This is all to say that it was extremely common for Roman slaveholders to abuse their enslaved women (and men and children) sexually.
Additionally, being "home-born" was the most common source of new slaves. Enslaved women were expected to give birth, which is often shown by clauses in bills of sale or wills that mention the transfer of ownership of an enslaved women along and any future children she may have. It was in the head of household's financial interest to abuse his slaves, but he was also at liberty to allow unofficial "marriages" between slaves, so sometimes he would encourage or arrange such relationships, too. Not all enslaved people born in a household would necessarily be the master's biological child.
However, an enslaved person had no father in the legal sense. The head of household could choose to expose a child born to an enslaved woman in his household, just as he could choose to expose an infant born to his wife. It was his choice whether to raise a child or not. Theoretically, he could also choose to raise an infant born to a slave as a legitimate heir, although this would have been very rare. While emotions could have affected these relationships in practice, biological children born to enslaved women were not legally or socially considered their children. They had no obligation to acknowledge paternity and old did so rarely, but historians do speculate that they may have been more likely the manumit such children.
If he chose to keep the child in the household, how that child was treated differed, just as how slaves were treated differently from household-to-household or based on their occupation. Some men seem to have had long-term sexual relationships with enslaved women who they would later free so that they could marry them (one ~charming~ document lists that the man will re-enslave his wife if she does not properly obey him, so the line between free and enslaved is often tenuous in these situations). In general, young men would be freed in their 20s and women after their childbearing years are over. Biological paternity certainly could affect these considerations, but since they were not "legally" the father, these sorts of relationships are not necessarily documented in the legal sources we have regarding sale or manumission.
Though Roman slaveholders may write about loving familial relationship they have with homeborn slaves, these slaves were not immune to being sold out of the household. And we don't have any accounts from the perspective of the slave. Depending on the person, maybe the biological child would be more likely to be kept, or maybe the holder would only consider the best economic options. First and foremost, the enslaved child was an economic asset. Though manumission did happen very often in ancient Rome, the majority of slaves were not freed. Basically, it it totally possible that some relationships were more parental or those considerations factored into manumission, but the sources we have are heavily biased toward the slaveholder perspective. It is likely similar to narratives from the American south that paint a rosy picture of slavery and relationships with slaves, in which we have much more evidence to show how inaccurate those depictions are.
As far as how the rest of the household viewed this type of situation, we have the most evidence for "jealous" wives, who beat and mistreat the enslaved people (both male and female) that their husbands had sexual relations with. In one account, well-born Roman women celebrated the holiday of Matralia by bringing in an enslaved women and ritually beating her as a sort of "scapegoat" for anger at husband actions. That is an extreme anecdote, but it's not hard to imagine a Roman matron, unable to do anything against her husband, abusing enslaved women and the children who look suspiciously like her husband.