Answer 1:
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 socalled 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 wellbehaved, 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 (socalled 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!

Answer 2:
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.

Answer 3:
Spacetime isn't really a substance that has
elastic properties like a normal fabric. The
curvature of spacetime 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.
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