I guess they all orbit around the solar system's center of mass (negligibly affected by the universal CoM), but that CoM probably moves around as the planets themselves move.

Relative to what, you might ask? That depends who you're asking 😉

All the solar system matter contributes to an object's orbital center but that's constantly moving as the system moves.

I think (?) most planets have their barycenter inside the sun's surface

The gravitational pull of system matter pales in comparison to the sun so you don't need to consider it for amateur purposes.

You can try KSP (Vanilla) versus Kopernicus mod if you want to feel the difference.

Also called n-body

The barycenter is different for each planet-sun (or any two object) pairing.

The earth and moon have a barycenter which is beneath the surface of earth. Likewise, the barycenter of the sun-earth pair is below the surface of the sun

Edit:

The barycenter of our solar system orbits the center of our galaxy (again in a barycentric manner)

Technically speaking, no celestial body in our solar system orbits around a single point. The barycenter thing only works with two bodies. When there are more than two bodies, such as in our solar system, the orbits become chaotic. Granted, the influence between planets is small, so they all appear to orbit their barycenters with the sun, but there are small perturbations to the orbits caused by the locations and masses of all the other bodies in the solar system.