Rousseau had five children by Therese le Vasseur, all of which they sent to an orphanage. Was this a common or acceptable practice at the time? Was it more common for unmarried couples?

Foundling hospitals (what we would now call orphanages) were usually populated by orphans, illegitimate children, and children of the very poor whose parents may try to reclaim them at a later date when the parents were better able to support a family. So, yes, it often happened that illegitimate children were sent away to a charitable institution or abandoned (or perhaps sent to a married family member who might treat the child as her own). It was still considered a shameful thing for most people to have a child out of wedlock (which was why, as evidence of "sinful behavior," illegitimate children might be sent away). Of course, these children might also serve as an impetus for their parents to marry.

In her book on illegitimate English children in the eighteenth century, Lisa Zunshine discusses that the problems that middle class women encountered with having an illegitimate child were not experienced by noble families (who had enough wealth to share some with a "bastard"), nor by poor couples who would often have common law marriages and not be particularly bothered by the fact that they weren't officially married. Poor single women were the worst off, and they were the ones who usually had to resort to giving up their children.

Rousseau, whose ideas about child rearing would become famous through his book Emile, was likely in a position to raise any children he may have fathered, but chose not to do so.