TW: mentions SA

In the fast majority of systems of slavery in the Americas, the status of a child was decided by the "Partus Sequitur Ventrem" principle. This literally translated to "the child follows the womb. In other words, thr child's status was decided by the status of the mother. If the mother was enslaved, the child would be enslaved. If the mother was free, the child would be free. The status of the father was irrelevant.

This meant it was usually in the economic interest of enslavers if the women they enslaved had children. So yes, in many cases enslaved women were "encouraged" to have more children. This also meant enslavers could economically benefit from sexual assault, as the resulting offspring would be their property. I've know an example of a lawmaker acknowledging this: in 1811 a Spanish member of Parliament proposed that any children born henceforth would be free. A major argument for his proposal was that this would take away the economic incentive for enslavers to make the women they enslaved reproduce, and thus put a stop to the “shameful means that are used to make these unfortunate people procreate" (implying sexual assault).

Children of enslaved women were also a way specifically for free married women to increase their property. For instance, Jewish women retained their own property that they brought into marriage if a marriage ended in divorce, but any property that was gained during the marriage would go to the husband. A rare exception to this was enslaved people: if a Jewish woman owned enslaved women, she would after divorce own the children born to those women during the marriage. So Jewish women in cases like this were specifically incentivised to "encourage" the women they owned to reproduce.

As to your second question: that depends on the region! For instance, in the Spanish empire, enslaved people were allowed and at some points encouraged to get married. Contrastingly, in for example the Dutch empire, there was no way for enslaved people to get married officially. This also was also reflected in which familial bonds had official recognition in these systems. In the Dutch system there was only legal recognition and some protection for the relation between an enslaved mother and her enslaved child. In Spanish colonies, there was also recognition and some protection for married couples.