Difference between Phonons and heat?

I will try to clear up your confusion step by step.

First of all: Your main question.

Phonons describe a collective excitation of atoms/molecules in solids. Think of vibrations in a material. They can be treated like particles which is why phonons are often called quasiparticles.

Heat is a form of energy which is an entirely different quantity. You are comparing apples and oranges when you ask whether phonons and heat are the same thing. Usually, phonons strongly influence the heat capacity in a solid but they are not identical.

Now, let's get to the sub-questions.

1. I answered that above.
2. They are not the same. Atoms in a lattice are vibrating, not the electrons or the bonds. "An entire medium vibrating" is the same thing as "its atoms vibrating". Phonons give rise to sound.
3. The fact that people use the omega symbol should not confuse you. There are too many quantities in physics, so sometimes the same greek letter is used for different quantities.
4. A dispersion relation does not have a unit. Quantities do.
5. Phonon frequency omega and wave number k are related via the dispersion relation.

Heat is energy stored in the (random) motion of particles. In a solid, the important particles are free electrons and the ions they leave behind. You could consider the motion of each ion separately, but it is usually more convenient to consider a collective motion of the ions at the same frequency. That's a phonon. That's also why there's an $omega$ in the dispersion relation. That $omega$ is an angular frequency --- the frequency the ions are oscillating at.

Phonons can be thought of as waves, and the dispersion relation tells you the relation between the wavevector (which can have units of $left[cmright]^{-1}$) of the wave and its frequency. In general, these relations can be quite complicated (especially compared to many waves in undergraduate physics, such as light in a vacuum, which has a linear dispersion relation $omega = c k$). A good starting point is to understand the dispersion relation for a one-dimensional lattice. (For example, see the wikipedia page for phonons.) It shows an example of a non-linear dispersion relation $omega propto sqrt{1 - cos{k a}}$, where $a$ is the spacing between the atoms in the lattice.

I mentioned (free) electrons earlier, and heat can also be stored in the random motion of the electrons. That can be quite important in metals, especially if you're interested in heat flow.

I suspect the difference between a phonon and heat wave is that the phonon wave length is always longer than the atom, and the wavelength of a heat wave is always the size or less than the particle. In essence the kinetic energy of heat is isolated to the particle, where sound wave is transmitted through a series of particles.