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.