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If a time machine were created, and we used it to steal a newborn Neanderthal, then give it to a volunteer to raise, what would become of it?

Could it learn English, and possibly even have children? Or would it eventually figure out it's not completely a human?

Well, let's starting quoting something from wikipedia:

The Neanderthal skull is typically more elongated and less globular than that of anatomically modern humans, and features a notable occipital bun.

Neanderthals are known for their large cranial capacity, which at 1,600 cm³ (98 cu in) is larger on average than that of modern humans.

Evidence suggests that Neanderthals walked upright much like modern humans.

Larger eye sockets and larger areas of the brain devoted to vision suggest that their eyesight may have been better than that of modern humans.

Neanderthals made stone tools, used fire, and were hunters.

However, rates of cranial trauma are not significantly different between Neanderthal and middle paleolithic Anatomically Modern Human samples. Both populations evidently cared for the injured and had some degree of medical knowledge.

The genomes of all non-Africans include portions that are of Neanderthal origin, a share estimated in 2014 to 1.5–2.1%.

Now, I'll answer your questions:

If a time machine were created, and we used it to steal a newborn Neanderthal, then give it to a volunteer to raise, what would become of it?

He/she would be raised as a otherwise normal boy or girl. Perhaps with some odd-looking traits, but normal otherwise.

Could it learn English?

Yes. And Spanish. Also French. Mathematics. Chemistry. Medicine. Computer programming. Economy. Chess. Driving vehicles. And so on...

Gosh, given the fact that neanderthals had a larger brain than modern humans, it is very possible that in fact they were more intelligent and smarter than us (or perhaps not). I think (but I might be wrong) that it is very likely that a median neanderthal would outperform a median modern human in many mental skills tests if both were to be raised in the same modern society and given similar education.

And possibly even have children?

They did.

Or would it eventually figure out it's not completely a human?

Stop being racist! Neanderthals are/were completely humans!

They can and in fact did interbred with modern humans, so this clearly shows that they and us are members of the same species.

We already have enough genetic, antropologic and archeologic evidence to firmly set that the Neanderthals are/were completely humans, whatever is a reasonable definition of "completely human". Saying that a Neanderthal is not completely human is not really different than saying that black people are not completely human. But, I don't want to accuse anyone here (including the OP, which probably was not aware of those facts). I just want to clear out the confusion and conscientize people to abandon the stereotype of seeing neanderthals as apes, subhumans, "not completely humans" or anything alike.

A Neanderthal might be a little funny-looking, but they were close enough to modern humans that you couldn't really tell. There are no physical or mental qualities shared by Neanderthals that do not exist in modern humans; the only differences are averages, and the Neanderthal average overlaps with the modern human average in all qualities.

Example: On average, Neanderthals had large noses compared to modern humans, but there are some humans around today with pretty big schnozes, so you couldn't prove they were a Neanderthal just by measuring the size of their nose. Same goes for their size, shape, voice, mental capabilities, and so on.

A Neanderthal in modern times would be able to get along just fine, though they might have a bit of difficulty getting a date. A scientist who happens to know the typical Neanderthal characteristics might be surprised by how much the guy walking down the street looks like a Neanderthal, but there would be enough doubt that they probably wouldn't mention it. Only a genetic test would prove there was something weird going on.

There may be some fertility issues between Neanderthals and modern humans, but not significantly more than what many couples experience today. Modern medicine may even make interbreeding more viable.

Strictly speaking. It would be obvious right away.

Much larger nose, shorter frame, altered pain sensitivity, and different vocal chords.

A Neanderthal would definitely be able to learn to understand English, at least the direct parts of it. It may even be able to speak it, though we would probably perceive an accent.

There may be some difficulty with the more ingenuitive activities that we take for granted, such as mathematics, cinema, or bridge design. Over the course of their reign in Europe Neanderthals did not appear to develop new technologies, or substantially change culturally. This might be explicable by the general difficulty of survival curtailing such developments, or by a genuine difference. There isn't a clear answer.

As for children, a Neanderthal would be compatible, particularly with those of European descent. Most Europeans are in fact hybrids of early Homo Sapiens and Neanderthals being about 3%ish Neanderthal. As to which Neanderthal sex is most compatible with modern humans is itself an open question.

TL;DR: In appearance, Neanderthals may look very different from "us", whoever "us" refers to, but then we look very different to each other.

Imagine in the pre-television era, the thoughts of a south-east-Asian peasant encountering Gerard Depardieu: "What is this strange, hulking creature?"

I suspect if we met a Neanderthal, we would not think him the most odd looking person we saw that month.

In terms of reproduction, Neanderthals could certainly breed with modern H. Sapiens, but it's not clear to what extent the offspring would be fertile. I suspect they would do just fine, but this is not actually known.

Long version

There is no bright line delimiting what differences between groups constitute them as two species, versus two subspecies.

Generally if two groups are incapable of mating and producing viable offspring, they are clearly separate species, but there is a considerable range of possibilities between species which cannot interbreed, and subspecies which are visibly different, and perhaps rarely interact, but are capable of interbreeding freely if the occasion arises.

For example, lions and tigers can interbreed and the offspring are generally fertile, though the second generation is often of delicate health. However they generally do not interbreed because their ranges do not overlap, and their habitat and lifestyle are different. So this is an example where the speciation process is almost complete.

The notion of "what is a species" is politicised, because the concept is used in conservation law. Therefore there are powerful incentives to classify two groups as separate species or as mere subspecies, in order to give an endangered group the protection of the law, or to deny it to them as a mere subspecies. These powerful social forces make it difficult to come up with any objective definition of the distinction between "species" and "subspecies".

It's not really known where in this range falls the distinction between Neanderthals and Homo Erectus.

• They are possibly simply a racial group, capable of interbreeding freely, producing offspring with a combination of characteristics from the parents.


• But it's also possible they are more akin to the Lion/Tiger example above, where interbreeding is possible, but the offspring are not necessarily particularly successful.


• If the offspring are partially fertile, then a female hybrid is likely to be more successful in reproducing than a male, based on Haldane's rule: https://en.wikipedia.org/wiki/Haldane%27s_rule
  

The answer to this question is not known.