Manganese has 7 valence electrons. While *filling* the orbitals you first give two electrons (spin paired) to the 4s orbital and then of the 5 electrons left, you give one to each orbital separately with the same spin in each orbital (Hund's rule). This gives you the pictures of the situation for part a.
Now the manganese in MnO2 has a +4 formal charge (O2 is -4 and for the molecule to be neutral Mn has to be +4). So go to your answer for part a and remove 4 electrons from the orbitals. In general, when *removing* electrons you want to remove from the 4s orbital first and then remove from the D orbitals, but Manganese is somewhat unique in that the 4s and 3d orbitals are so close in energy that it doesn't matter as much (the answer for part d assumes that you removed the 4s electrons first to form the ion). No matter which 4 electrons you remove you are going to be left with an unpaired electron in some orbital and that makes the Mn+4 ion paramagnetic (in order to be diamagnetic, all of the orbitals would have to have 2 electrons in them so that all the electrons are spin paired to cancel out the paramagnetism...anything with an unpaired electron left over in any one of its orbitals is considered paramagnetic).