My wave function and pchem is very weak, but the second statement about the s and d orbitals and the p orbital refers to inversion symmetry. If you take any point and then follow it through the center of the atom, to the other side, that is an inversion point. For the s and d orbitals, if you take any point and invert it through the center, then you get the same sign on the orbital and the same symmetric point. For the p orbital, it is symmetric with respect to shape, but the orbital sign is opposite, so it is anti symmetric.
Sorry I can't help with the other questions.