Nitrogen, phosphorous, antimony, and bismuth are all in the same group (column) of the periodic table.

The atomic radius increases from the top of a group to the bottom, due to increased principle shell number (n). As one travels down a group, another s shell is added, meaning that electrons are added in another orbit farther from the nucleus. This serves to increase the atomic radius of the atom.

Lithium, boron, oxygen, and neon are all in the same row (period) of the periodic table.

The atomic radius decreases from left to right along a period due to increased effective nuclear force. From left to right the atomic number increases, indicating that more protons are added. The addition of protons increases the positive charge in the nucleus, pulling in the outer electrons by increasing the effective nuclear force, decreasing the radius.

In math terms, we can equate effective nuclear force using the force equation between two charged particles.

We can see that the farther apart the electrons and protons are, the less the force is between them.

The second ionization energy of a neutral atom is always greater than the first.

Recall that ionization energy refers to the energy required to remove an electron from an atom. The first ionization energy will result in a cation. The reduced number of electrons in a cation allows them to be pulled closer to the nucleus by the positively charged protons, effectively decreasing the atomic radius and increasing the attractive force between the electrons and the nucleus. The energy required to remove a second electron must overcome this increased affinity, and will thus be greater than the first ionization energy.

The rest of the answers can be examined with a few simple trends kept in mind. Atomic radius increases to the left of a period and down a group. If two ions have the same electron configuration (like the chlorine anion and potassium cation), then the anion will have a larger radius because it has fewer protons to draw electrons inward. Electronegativity and electron affinity increase to the right across a period and up a group.

Atomic radius increases down each group of the periodic table and toward the left of each period. Since the elements listed are all in the same group, iodine would have the greatest atomic radius because it farther down the period compared to the others. 

Energy level increases moving down a group of the periodic table. As energy level increases, the outer valence shell becomes more distant from the nucleus, causing atomic radius to increase.

Energy level remains constant across a period, but electrons are added within the same orbitals. When new electrons are added within the same orbital, additional protons are also added to the nucleus. This increases the effective nuclear charge, pulling the electrons closer to the nucleus. The trend for atomic radius is to decrease as we move right along a row.

This means that the general trend for atomic radius is to increase as one moves to the left and downward on the periodic table.

Chlorine has a great thermodynamic desire to capture an electron, thus taking on the electronic structure of a stable noble gas. This causes chlorine to release energy when it captures an electron as it becomes more stable.

Sodium, on the other hand, would prefer to lose an electron and gain the configuration of a noble gas. Adding an electron would however award some stability to sodium, due to the complete s orbital that this would ensue.

Second electron affinity is usually encountered for such elements as oxygen and sulfur, which form anions with the addition of two electrons. The first electron affinity gives you O^- or S^-, and so it takes significant energy to add another electron to an already negative ion.