Note: if this answer is too long, you can skip to the main conclusion section below.
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 Mozart at an early age.
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. "
Harry Harlow 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):
... Harlow 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. Harlow next chose to investigate if the
infants had a preference for bare-wire mothers or cloth-covered
mothers. 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. Harlow
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 Bowlby's
assertions on the importance of love and mother-child
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.
Wile Harry Harlow's 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. You would therefore need to incorporate sensors for body heat, a layer of soft material designed to be pleasant to the touch on the outside, and an internal source of heat.
But importantly, it appears that robots need not look human. Feral children recognise animal caretakers as valid caregivers and will accept their care, a fact born out simply by the surprisingly large number of feral children out there. It is, however, in the children's best interests to have robots mimic humans physiologically, on the premise that substantial differences in physiology can lead children to imitate their caregivers and try (for example) to walk differently than regular humans - this is also born out simply by looking at the histories of feral children.
Note that this section doesn't talk about, for example, toys, playthings or other ways to keep children occupied - only what is needed on the robot's part to facilitate social ability. I presume the environment will take care of providing toys and opportunities for play - humans need to be kept entertained, after all, and this falls into the level of common sense that I presume the environment will provide.
tl;dr Robots don't have to do anything to force motor skills to develop. Note: see concluding paragraph of last section for note about presence of toys.
Surprisingly, motor development is one of those things that babies seem to learn pretty much on their own. All babies learn how to climb stairs by the same age, regardless of the presence of stairs in their house. While there is evidence to suggest that cultural differences can stimulate onset of motor development in different ways, it is difficult to find evidence that motor development itself is something that actually requires an active agent.
Since it appears that all babies need for motor skills is an actual environment to explore, we can assume that this is subsumed in the environment and nothing is needed on the robots' part.
tl;dr It's not clear whether robots actually need to have facial expressions to help facilitate picking up on nonverbal cues. I suspect it is not necessary if children are able to play with each other on a regular basis.
There is only one notable and possible exception to the conclusion reached in the section on motor skills: the development of facial muscle control - or oral motor skills in the literature - is still poorly understood.
According to R. S. Feldman in his book who in turn cites authors Freedman, Charlesworth and Eibl-Eibesfeldt who conducted a series of experiments in the 1970s, facial expressions between congenitally blind and sighted children develop in the same way - they are innate. What we don't know, however, is to what extent nonverbal communication is acquired - Feldman cites evidence to suggest that it is more innate than acquired using cross-cultural studies, but ultimately admits this isn't as well-explored as we would like.
Children respond to nonverbal cues in different ways at different ages. Four-month-olds, for example, are "more expressive when their mothers are facially active" and can "respond to bizarre deviations from normal facial expressions such as length 'deadpan' expressions". One-year-olds, on the other hand, are more acutely aware of facial expressions and can rely on the facial expressions of their mothers when placed in a position of uncertainty to make a decision.
It isn't clear if it is necessary to incorporate the ability to provide facial emotions into our robots to encourage nonverbal communication. It would help, obviously, but as I've stated, we're only interested in what is required on the robot's part to produce this.
Based on the limited evidence I have, I am not convinced that our robots need to exhibit facial emotions. Children are sensitive to verbal cues (such as pitch and tone) as well, after all, and I suspect that encouraging play between children will more than allow them to develop the finer aspects of this art. We have already removed children with conditions known to delay or adversely affect development from our group, so this should be fairly inclusive overall.
tl;dr Robots do need to speak and be able to speak like normal human beings. They need to be programmed to speak to children frequently and often, and they need to be able to communicate contextual warnings using pitch and tone.
Linguists universally agree that all children learn languages in the same way, regardless of the language or dialect used. Children will learn the language spoken around them, according to Indiana University's Linguistics Department - deaf children will even pick up sign language if exposed to it.
Speech and language are separate creatures. Children are capable of speech as their motor development continues to grow, while language involves the articulation of valid sentences. Languages are learned gradually, mostly through imitation but potentially aided by correction and frequent engagement.
Therefore, any artificial robot or environment must expose children to coherent and articulate language. This requires, at the least, the ability to handle complex natural language processing, the ability to provide context by tone and pitch, and a solution to the cocktail party effect if robots are handling multiple children. At some point, obviously, robots will need to be able to respond to children's questions meaningfully, so this goes without saying.
Higher-Level Processes, or The Murky Area of Parenting
tl;dr Your robots would need some mechanism of resolving ethical dilemmas, and a means enforcing authority within the context of prescribing right behaviour without permanently damaging the children's ability to think independently, so that they can teach children who are willing to learn. This is really difficult - possibly nigh impossible - and an alternative remote human / robot on-site solution is advocated.
Now we come to the more difficult aspect of raising children: how should our children and robots embrace the concept of authority? This is relevant: in the long term, authority is what gets children to do what they don't want to do, including learning new things.
A warning: I wish I could still offer peer-reviewed studies from this point onwards, but this is now simply too big a field to identify definitive consensus. This is the part where we make educated guesses and draw on our natural instincts of future or current parents.
First, it is absolutely imperative that some notion of authority is put in play. Our children need to be potty-trained and will not do so of their own accord, for example, while our robots need to be able to handle defiant behaviour (as parents of toddlers in the Terrible Twos learn).
At the age where children can conceive of such abstractions, one obvious model is to offer punishment/reward schemes to facilitate good and bad behaviours. This model was in vogue during the behaviourist paradigm of psychology - however, a bird's eye view of recent thought in this regard tends to favour the idea that recognising positive behaviour is more important than punishing negative behaviour, and has long-term benefits in terms of emotional stability.
For this to happen, though, the robot needs to be able to provide appropriate recognition and more importantly recognise positive behaviour. I can think of two ways for the former to happen: either appreciative tones are used, and/or appropriate treats (a particularly tasty meal, permission to enjoy extended periods of time for playing, etc.) can be provided. The former provides children intrinsic incentive; the latter reinforces the notion that the robots provide value. You can even teach everything this way: people learn to fold clothes or math or advanced science by being given lessons and earning the robots' approval, as well as by allowing robots to encourage interest in a subject.
What I don't know - and what I suspect is a fairly remarkable challenge - is how your robots can recognise what constitutes good behaviour.
This is difficult: does standing up to a bully and injuring him in the process constitute good or bad behaviour? Your robots would need to have some mechanism of correctly defusing the situation, as well as realising who was at fault, and dispensing adequate justice. This is a task so difficult not even humans can manage it perfectly, as all of us can attest to at some point.
One very naive way to do it is to recognise conflict as behaviour that impedes a specific goal, and recognise good behaviour as classes of behaviour that advance said goal. Fighting, for example, is Bad because it impedes the goal of harmoniously working together. A robot could identify such behaviour and nip it in the bud at once. Performing an activity that allows another child to perform well is Good, and a robot can offer such assistance at once. (I presume computer vision has advanced to the point that robots can recognise these activities in the first place).
The problem with this is that it ignores underlying reasoning or motivation. Standing up to a bully can be argued to be good, even though it encourages Bad behaviour, as the reasoning - as a show of resilience - could justify it as being good. So what should the robot do? How does a computing machine evaluate arguments?
I'll go off on a tangent for a bit and point out that the recent trend of machine learning and big data is actually harmful to artificial intelligence approaches precisely because it is no longer sexy to teach AI how to reason, rather than make decisions based on patterns extracted from data. There has been work on this in the past - computers can apply logic to solve syllogisms if the rules are coded in the past, and probabilistic reasoning is a growing field - but if we one day want AI that is able to actually demonstrate critical thought and reasoning, more work needs to be done on stuff like, say, second-order logic.
So there you have it, the fatal flaw. Unless robots can be taught how to evaluate reasoning and arguments, it is perfectly useless at raising children. The only way out of this is some sort deus ex machina: a knowledge pill, for example, that instantly teaches kids how to do this or that.
Either that or you have an actual human watching the facility from afar, recognising good behaviour, providing emotional counseling, and in general doing all the things a robot can't do when the children reach past a certain age. I think this is the best possible compromise: have an actual human supervise kids through teleconferencing and use robots as 'arms' after the kids hit a certain age.
The robots raising these children, at a bare minimum, need to be able to possess advanced language interpretation skills, human-like skin with appropriate thermal regulation and somewhat humanoid physiology, but don't really need to be able to mimic human facial expressions or even look all that human in terms of overall design. It's assumed that the environment and the robots are two separate entities: the environment includes a supply chain of relevant necessities, the robots are in charge of distributing it. Think of the environment as a store, and the robots as caretakers who sample from the store.
Based on the reasoning above, it is possible to have robots that can provide neonatal care, but not robots that can replace the role of teachers as children grow older unless AI becomes capable of evaluating arguments at a near-human level. (And then what's stopping a robot revolution? But I digress).
This makes it impossible for a fully automated farm to take care of children until they are 14, but a suitable mix of humans supervising remotely and robots on site can help resolve many of these issues and make the whole endeavour possible. I think this is actually the best way possible.
All of this is conditional on the assumption that people use common sense when handling babies (like making sure they have ample things to play with, have room to explore, are taken care of in terms of diaper changing and food, as well as are in well-lit rooms in moderate temperature conditions) and that children have the opportunity to regularly play with each other.
That should be more than enough for such an experiment to succeed.
Disclaimer: I am not a trained psychologist or doctor, all of the above conclusions are my own and should not be taken as advice, all of the above research is fairly limited and only barely scratches the surface, so don't think it is perfect or complete because it really isn't. Some links may be behind a paywall.