Space is a mathematical abstraction. You can do anything with, because it's just math. If you want to bend, curve, distort, rip space, or add more dimensions — go for it.
Physical medium should be compressible, like any other medium. Higgs «field» (let's call it Higgium — Higgs+Vacuum) is presented everywhere, because Higgs boson gives mass to every particle in the Visible Universe and beyond, so it should conduct gravitational waves. As demonstrated by LIGO/VIRGO, gravitational waves causes distortions in light travel. These distortions can be explained as changes of conductivity in Higgium, which can be caused by changes in density, so yes, it can be waves of compression.
Spin-0 particles are round particles, like o (like ball), or something that cannot rotate at all. Spin-2 particles are symmetrical particles, like 8 (like H2 molecule).
For spin-0, it can be changes in density only. For spin-2, it can be changes in orientation, in rotation speed, in rotation orientation, Dzhanibekov effect, wobbling, and some other if it has more complex shape, e.g. two spirals connected.
The geodesic distance between inertial test masses increases and decreases.
Colloquially we say that space has stretched and compressed.
Geodesic distance can be measured by the time it takes light to travel between the points.
Inertial test masses can be simulated by hanging them from fine wires (LIGO) or by setting them up in an orbital constellation (LISA).