I am not sure into how much detail I should get into here, but this is not quite right
In Wales:

They move so fast that you end up with a 3d representation of the orbital shape that looks solid. But in reality if you could freeze at any one moment, there would be dots of electrons in random looking places around the nucleus.

Electrons don't have trajectories, and as you know from molecular bond theory there is such a thing as orbital hybridization which is just a change of basis achieved using linear algebra. For instance, you start with one (spherical) 2s orbital and three 2p orbitals (aligned along the X, Y, Z axes) and you end up with 4 orbitals pointing at the vertices of a tetrahedron. The point is that the decomposition is mostly arbitrary. In particular, there's no reason to choose one collection of five orthogonal 3d orbitals over any other unless there's an external electromagnetic field.

Or we think we are. We are nothing more than a probability.   Each atom is full of empty space.  How am I so solid when I am so full of space?  Why does matter not sift through me as though I were a ghost?

Have you studied the basic Born-Oppenheimer theory of the hydrogen molecule? Have you looked at how the atomic orbitals hybridize into binding and non-binding molecular orbitals? If you have the two electrons in a non-binding molecular state you can see how the two atoms bounce off each other - that is, they behave as if they're solid. It's all a consequence of combining the Pauli exclusion principle and quantum-mechanical superposition of states.

I've been neutron scattering.  I've fired streams of neutrons at my samples and collected the scattered hits to tell me what shape my molecules are. The neutrons go into my samples and bounce off from inside the atoms and the angles in the scattering patterns tell me the size and shape.

Yes, because the neutrons are not charged and so they don't really interact with the electron clouds, only with the atomic nuclei. Well, there is a small scattering cross-section of electrons by neutrons, but if the neutrons are energetic enough the most they'll do is knock an electron off the molecule and go on on their merry way. But photons see the electron cloud because charged particles interact strongly with them.

Photons are both waves and particles.

"Things propagate as waves but interact as particles" would be a possible way of saying this.

Particles goes through me, so am I really solid? Am I really here?

You have a small interaction cross-section with certain particles. Just because a glass pane is transparent doesn't mean it's not solid, isn't it? Now, depending on your definition of solid, you're most emphatically not. Parts of you are liquid, parts of you are gels or emulsions, parts of you are rather elastic. Only your bones and teeth are solid and not througout their whole volume.

We have met the enemy, and he is us — Pogo