(iii) and (iv) are similar. I will do (iii) and leave (iv) for you to do. For some reason I got directly the opposite of what you said. Are you sure you are right?(iii) S(t) is even iff F(-omega) = F(omega)
(iv) S(t) is odd iff F(-omega) = -F(omega)
(Note the converse is basically the same idea).
EDIT I just realized why I made a mistake. When I made a variable substitution s=-t I forgot to attach the minus sign in front of the integral. I am just lazy to retype the entire thing. So you are right.