you see that both the red and the black sinosodial curve represent the "same" physical state (the particle going up and down). They carry the same information.

Since k are multipiles of reciprocal lattice vectors, we see that each k of this kind carry the same information as the smallest one, i.e we only need the k that is contained in the 1st BZ. The other ones, we can "translate back" with a multiple of reciprocal lattice vectors.

Hmm yes, I think so. You get an infinte set of soultion, but you can always relate them to the smallest one (smallest magnitude), by substracting or adding reciprocal lattice vectors.

Why does that same argument not imply that electromagnetic waves with wavevectors outside of some zone do not carry extra information? I guess I am confused about how the periodicity of the reciprocal lattice makes that argument true? If you put more of those black dots in there, would they just go up and down also, or is there something special about the black dots that they chose?