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We can explore what sort of robots we would need by looking at the absolutely critical factors that children need to grow into mature adults with the core behaviours you mentioned, and whether/how robots can facilitate the same.
There is no magic bullet to this: no collection of robots can absolutely raise a perfect group of children. There are genetic, congenital and even accidental factors involved that can strongly impact lifetime development that you can't necessarily weed out completely. Nor can you expect a thoroughly perfect answer: children's development is still very much an active research area, and there are few certainties in this regard.
Assuming only that a select group of children are introduced into this artificial environment that meet a minimum bar (no damage, no genetic diseases, etc.), then we can proceed on that assumption to flesh out what factors are needed to provide future care.
What I'm Going to Ignore: For the purposes of this question, we will assume a certain level of common sense e.g. that actively dangerous substances or materials are prohibited in the child rearing environment, that it meets a minimum standard of human care (e.g. no extreme sensory deprivation, it is light, not too hot, not too cold, etc.), and that some care is taken for granted (i.e. age-appropriate consumption and regular diaper changing are enforced, but everything else is not guaranteed).
All we want is to know what the bare minimum an active agent needs to provide to stimulate the relevant skills - in other words, we will also ignore needless frills that encourage faster development but are not required to stimulate the necessary skills, like, say, making them listen to <PERSON> at an early age.
Sensory Stimulation
tl;dr Robots need to have human-like skin but don't have to look human, though it is best if they do.
There is some evidence that suggests that sensory stimulation between mother and child between an active agent and children plays an important role in infant's long-term social development. Foals exposed to limited maternal contact post-partum experience "patterns of insecure attachment to their mothers (strong dependence on their mothers, little play) and impaired social competences (social withdrawal, aggressiveness) at all ages. "
<PERSON> explored the mother-child bond among rhesus monkeys. Part of his experiments involved rigging up inanimate surrogate mothers (excerpts taken from his Wikipedia page, with points for citation removed except where marked as needed):
... <PERSON> created inanimate surrogate mothers for the rhesus infants from wire and wood. Each infant became attached to its particular mother, recognizing its unique face and preferring it above all others.[citation needed] <PERSON> next chose to investigate if the infants had a preference for bare-wire mothers or cloth-covered mothers. 693
For this experiment, he presented the infants with a clothed mother and a wire mother under two conditions. In one situation, the wire mother held a bottle with food, and the cloth mother held no food. In the other situation, the cloth mother held the bottle, and the wire mother had nothing.
Overwhelmingly, the infant macaques preferred spending their time clinging to the cloth mother. Even when only the wire mother could provide nourishment, the monkeys visited her only to feed. <PERSON> concluded that there was much more to the mother-infant relationship than milk, and that this "contact comfort" was essential to the psychological development and health of infant monkeys and children. It was this research that gave strong, empirical support to <PERSON>'s assertions on the importance of love and mother-child interaction.[citation needed]
Successive experiments concluded that infants used the surrogate as a base for exploration, and a source of comfort and protection in novel and even frightening situations. In an experiment called the "open-field test," an infant was placed in a novel environment with novel objects. When the infant’s surrogate mother was present, it clung to her, but then began venturing off to explore. If frightened, the infant ran back to the surrogate mother and clung to her for a time before venturing out again. Without the surrogate mother's presence, the monkeys were paralyzed with fear, huddling in a ball and sucking their thumbs.
<PERSON> later experiments are largely considered unethical (he subjected rhesus infants to extreme long-term sensory deprivation for up to two years, with no light in a confined environment), it would appear that having robots that mimic the sensation of skin are in some ways required for children to be socially well-adjusted later on.
Such robots would be best off mimicking the characteristics of maternal skin as much as possible. For example, new mothers' skin is responsive to their babies' temperature. 362