Did you want to know something specific or are you just inviting comment? It is important to say.
I think you have worked harder than you needed to.
Definition: A shape of form ##y=f(z)## traveling in the +z direction with speed ##v## has form ##y(z,t)=f(z-vt)##
... this will be a traveling wave if it also satisfies the wave equation. (Do all such functions satisfy the wave equation?)
... a definition like that allows positive values of v to mean that the waveform propagates in the positive z direction - making it easier to keep track of minus signs.
For (a): ##y(z,t)=A\sin^2 4\pi(t+z)## ... this is a traveling wave with form ##f(z)=A\sin^2 4\pi z##
This means that ##z-vt = t+z \implies v=1\text{ (unit)} ## ... i.e. the wave propagates in the +z direction.
See how that somes easily?
There is also no need to go into wave numbers and angular frequencies.
You don't need the ##\pm## sign in your definitions unless you insist that the constants ##\omega## and ##k## can only take positive values.
Fortunately you don't have to prove that (c) is not a traveling wave.