Let's start with your second question. First, I assume you're only talking about classical gravity - if we include quantum mechanics in the mix, there's no reason to expect spacetime to be "smooth" at very small distances, and in fact, spacetime might even cease to exist completely! But you're right that for usual classical gravity, spacetime is smooth (as long as the matter in it is smooth as well). The reason for this is the same for which most physical systems are "smooth," in some sense: the equations that describe them require them to be smooth!

More technically, physics is usually formulated in terms of so-called differential equations: these are equations that describe how something (for example, the velocity of water in a pipe, the temperature in the atmosphere, the curvature of spacetime, etc.) varies in time and space. In order for these equations to be well-behaved, the solutions need to have certain smoothness properties (with certain exceptions when things go "bad," like when the water in the pipe hits a wall, or at the singularity inside a black hole where the curvature of spacetime blow up, etc.). So, that's what keeps spacetime smooth.

Now, let's go back to your first question: what allows spacetime to move back to how it was before matter influenced it? Well, it's the same thing: the differential equations describe how spacetime reacts to the presence of matter, and therefore changing the distribution of matter requires spacetime changing! Let me point out, though, that removing matter from a spacetime doesn't necessarily require it to go exactly back to the way it was before. For example, say I start with empty, flat space. Then I put a blob of matter in the middle (say, a star). Obviously, spacetime will curve in response to the matter. Now, say I then make the matter disappear (nevermind how!). The spacetime will try to "spring back" to flat space, but it won't become perfectly flat - there'll be residual curvature that moves around in the spacetime (so-called gravitational waves). So the spacetime will still know about the matter that was in it, in a way, even if it's not there anymore.

I hope this helps answer your question a bit!

That's a very interesting question. It turns out that spacetime doesn't just curve, it also stretches twists, bounces, and sometimes even rips apart. All of this interesting behavior is described by an equation know as the Einstein equation. It's a little difficult to explain precisely because the formulation of the Einstein equation relies heavily on a branch of Mathematics called differential geometry. However, you can get a long way by thinking of spacetime as being like the surface of a lake. Matter moving through spacetime is then kind of like a duck swimming across a lake. As the duck moves it not only curves the surface of the lake, it also makes ripples as the surface settles back down to being flat. The same thing happens in gravity and these ripples are called gravitational waves.

I should mention that while physicists have seen indirect evidence that gravity waves exist, these waves have never been directly measured. The problem is that gravity waves tend to be very, very tiny. However, when two black holes merge in our galaxy the gravitational waves are just big enough that it may be possible to build a detector on Earth to measure them. The experimentalist working on this project think that their detectors will be sensitive enough to see gravitational waves sometime in the next few years and when they do it will be a very important discovery. If you're interested you can read a lot more about it here.

Space-time isn't really a substance that has elastic properties like a normal fabric. The curvature of space-time and the mass contained in it are identical.

I know that a lot of physicists see problems with this description. Like the Newtonian theory of gravity before it, general relativistic gravitation is probably an approximation of what is actually going on, but we don't know what the reality is yet.