Does matter follow the curvature of space-time?

Questions of this sort in relativity are quite dodgy, because the notion of "size" is inherently subjective--literally, it is dependent on the observer in spacetime. Even in special relativity, we know the phenomenon of length contraction. A fast-moving box will appear to a stationary observer to be shorter, but in its frame it will be the same dimensions. This is slightly altered in general relativity, but the observer-dependent nature of the question remains.

You can think of spacetime curvature as a change in your spacetime yardstick as you move to different points in spacetime. Say, per your question, we have a box in the presence of a black hole. We drop the box, and it falls in. Like the apocryphal Alice or Bob, from our perspective the box will spaghettify--it will appear to be stretched, contorted, due to the effect of gravity on the light it sends our direction. In its frame, it sits in freefall. Indeed, as it approaches the black hole, the gravity at its bottom will be greater than the gravity at the top. This will induce some stress within the box. If the box is conventionally sized by our standards, in almost no observable astrophysical black hole will this become an issue for the box--the tidal disparity due to gravity is too small outside the event horizon. Similarly, a box in a rotating spacetime may experience some tidal shear force resulting from "frame dragging" of the gravitational field. All these effects would be miniscule, except in a region of extremely high spacetime curvature.

But in principle, an object with any finite extent will be subject to the changing metric on spacetime in the presence of a gravitational field. The notion of "length" simply ceases to be very helpful when your measurement of such a quantity depends on a dynamical field.

How is matter affected by the warping of space-time?

To start with, if there is not matter in the universe, there is no warping. Everything is flat.

If the only matter in the universe is your "test matter" , it will be the one warping spacetime around it, as three dimensional space goes, it will just be the newtonian gravitational field, unless it is a huge mass moving with relativistic velocities when one has to learn the mathematics of general relativity to plot the way a single mass warps space-time.

If we are in the present universe where for large masses and large velocities general relativity has to be used , there is spacetime curvature defined by all those masses in the universe, in the (x,y,z,t) point where the center of mass of your test "mass" is.

Does it expand and contract,

Not for low values of energy and mass of your test particle. In the regime where general relativity has to be used and the curvature is strong, there are various effective forces acting on the masses .

follow the curvature of space?

If it is a test point mass, it will follow the curvature of spacetime. Other wise its own mass will also contribute to the curvature.

What happens the shape/volume/density of matter when it enters a gravity well,

a gravity well belongs to Newtonian gravitation. When at relativistic sizes and speeds you can see what happens in the presence of large masses in the LIGO experiment.

or washed over by gravitational waves?

again you can get an idea from the LIGO obsrvation of the merging of black holes.