Is new space(time?) being created as the Universe expands, or does the existing spacetime just get stretched?
If it just gets stretched, why do galaxies move along with the expansion instead of just getting smeared (like a drawing on an inflatable balloon)?
Since the second part of your question is a duplicate I'll address just the first part. however I suspect you're going to be disappointed because my answer is that your question doesn't have an answer.
The problem is that spacetime isn't an object and isn't being stretched. We're all used to seeing spacetime modelled as a rubber sheet, but while this can be a useful analogy for beginners it's misleading if you stretch it too far. In general relativity spacetime is a mathematical structure not a physical object. It's a combination of a manifold and a metric. At the risk of oversimplifying, the manifold determines the dimensionality and the topology, and the metric allows us to calculate distances.
We normally approximate our universe with the FLRW metric, and one of the features of this metric is that it's time dependant. Specifically, if we use the metric to calculate the distance between two comoving points we find that the distance we calculate increases with time. This is why we say the universe is expanding. However nothing is being stretched or created in anything like the usual meaning of those words.
Short answer: Unlike stretchy materials, spacetime lacks a measure of stretchedness.
Long answer: Let's see where the rubber sheet analogy holds and where it fails. When you stretch a rubber sheet, two points on it change distance. You can also formulate a differential version of this property by saying it about infinitesimally close points. Similar is happening for the spacetime. In fact, description how infinitesimally close points change their (infinitesimal) distance is a complete geometric description of a spacetime in general relativity (GR). Even more, it is complete description of a spacetime in GR (up to details about the matter content in it).
This is described by a quantity called metric tensor $g_{\mu\nu}$, or simply metric for short. It is $n$ by $n$ matrix, where $n$ is the number of spacetime dimensions (so it is usually 4 unless you are dealing with theories with different number of dimensions). Square of the infinitesimal distance is given by: \begin{equation} ds^2=\sum_\mu \sum_\nu g_{\mu\nu}dx^\mu dx^\nu \end{equation}
In Einstein summation convention, summation over repeated indices is assumed, so this is usually written simply as $ds^2=g_{\mu\nu}dx^\mu dx^\nu$.
For flat spacetime $g_{\mu\nu}$ is a diagonal matrix with -1 followed by 1s or 1 followed by -1s, depending on the convention (it gives the same physics in the end). For a flat Euclidean space, metric tensor is simply a unit matrix, so for 3-dimensional space this gives: \begin{equation} ds^2=dx^2+dy^2+dz^2 \end{equation} which is just a differential expression for Pythagorean theorem.
Where does the analogy fail?
Well, as I said before, metric is a complete description of the spacetime in GR. This means that the spacetime doesn't have any additional property. Rubber, on the other hand, has a measure of how much it has been stretched. This is closely related to rubber having a measure amount of rubber per unit space. But there is no such thing as "amount of space per unit space", so once you stretch the spacetime, it won't go back. Or you could say that stretching and creating spacetime is the same thing, because you end up with more volume "for free" (meaning that, unlike rubber, it has no way to "remember" it used to have less volume).
Expansion of space means simply that distances between the points become larger, without any movement implied. It is simpler that with stretchy materials, but less intuitive, because our intuition forces us to imply additional properties which are not there.
If the stretching analogy were to be more than that, and was something actually happening to expanding spacetime, that would result in an increase either in elastic potential energy or elastic tension. That increasing tension would result in an increase of the speed of gravitational waves and a reduction in the effective mass of objects, none of which have been observed
The expansion of the universe is just the relative motion of objects in it. There's no definable difference in general relativity between ordinary relative motion of two objects and those objects being "stationary while space expands between them". At best, they're descriptions of the same phenomenon in different coordinates.
Some sources that talk about expanding space are just using it as alternate description of the same physics. This isn't strictly wrong but I think it's unnecessarily confusing. It leads to questions like yours which would probably never arise if people understood that the motion of galactic superclusters is the same as any other motion.
Some sources explicitly say that the expansion of the universe is different from other kinds of motion because it's due to the expansion of space itself. They're just plain wrong, and you should be suspicious of everything else they say about cosmology.
