Fermions have spin 1/2, so matter cannot all be in the same place and we cannot walk through walls
Correct. This is the Pauli exclusion principle, which says that for fermions only 2*spin +1 (= number of spin states (all these states are quantum states) = number of different internal states) particles can be in any external state (given by momentum, place in space).
Photons have spin 1, so like electrical charges repel and opposites attract
Gravitons have spin 2, so like matter attracts gravitationally, like being like all matter we have found
These are not at all comparable to the previous statement. The analog for the statement about fermions is that bosons prefer to be in the same state. For low energies this leads to Bose-Einstein condensation.
In my opinion the easiest explanation of the different particles' spin are the following:
The photon having spin 1 is analogous to saying that an interaction with one photon changes the spin by 1 unit, or that it is a dipole interaction. (second order multipole expansion. The first order is a monopole which is equivalent to a charge. Since the photon is to leading order a dipole it can not carry electric charge.) The electromagnetic interaction being a dipole interaction to first order can be shown in classical electrodynamics.
Aside - the multipole expansion may be most familiar to you from electrodynamics or especially different radio modes (the simplest radio receiver is only sensitive to dipole fields and is thus frequently called a "dipole"). It is basically a decomposition into spherical harmonics.
Typically gravitons having spin 2 is stated as a consequence of the waves in linearized gravity (approximately the gravitational waves we measure today. Also linearized, because that makes the equations much simpler and the theory of general relativity in its least approximate form is extremely complicated.) being quadrupole waves to leading order (third order multipole expansion), as this is an easier calculation. A quadrupole interaction changes the spin by 2 units. This is analogous to an interaction with two photons.
Now, about being a repelling or attracting force:
she references the Feynman lectures.
... and a quick look into the "Feynman Lectures on Gravitation" (on archive.org) reveals that this is in fact only a statement in this book and not a calculation. Take a look on page 30. Maybe there is something else, and maybe I will look further.
Assuming these statements to be true, I would change your statements 2 and 3 to the following:
- Since in electromagnetism like charges repel, the associated mediator particle needs to have odd spin.
- Since in gravitation like charges attract, the associated mediator particle needs to have even spin.
Note that this does not fix the spin. I made this change to better represent the direction of the reasoning: We start with an interaction with a known property (like charges attract/repel) and want to deduce the properties of an associated quantum theory (spin of the mediator particle).
Lastly, let me state that the existence (and properties) of the graviton as a particle relies on it being possible to formulate Gravity as a quantum theory, which has not yet been done and has proven to be a physically difficult problem.
Tl;dr:
Yes to all three, but 1 is completely different from 2 and 3.
