This is a very interesting question and we will need to get into some detail to answer it, but I think it will be worth the try.

Here is a short answer: Because the change in the atomic number implies an important change in the structure of the elements, and the forces between the atoms are therefore very different, when the forces are huge you will have them solid and when they are much less intense you will have them as gases. But if you want to know the details of this situation, read on.

Let's begin by rephrasing your question. Even if I perfectly understand what you meant by "some elements are gas and others are solid", I hope you will agree with me in adding "at room temperature", because if you make the temperature high enough, you will be able to liquefy most and vaporize all of the elements at ambient pressure. So, in other words your question could be stated: why the melting and boiling point of the chemical elements varies so drastically as we move along a the first period in the periodic table? As you have correctly pointed out the next neighbor has only one more proton... and that makes a huge difference!!!

Taking the example of C and N, (this statement can be generalized) I can say that even one proton and electron more in the atom can radically change the way the atoms can establish links among themselves (in chemical terms: form bonds) to show the different forms of pure elements at room temperature. We have to go now down to the details of the electronic structure so that you can better understand. Carbon has an atomic number of 6, that means 6 protons in the nucleus and six electrons orbiting around it. Those electrons are distributed into the so called "electron shells". The six electrons of carbon go this way: two in the first shell and four in the second shell. We call those in the second shell "core electrons" and that means they are internal in the electron structure and not ready to participate in the bonding of the atoms. So we have four left. So the most stable form of carbon is that where every carbon atom is surrounded by other four carbon atoms and each pair is sharing two electrons. In this ways all of the atoms can count eight electrons in the second shell and that is a very stable electronic configuration. The solid thus formed is called diamond and since the carbon-carbon covalent bonds are very strong and each atom is linked to four other atoms, the whole crystal is extremely stable and can be heated in the absence of air up to 3825 degree Celsius before it vaporizes.

Going to N (nitrogen), now the outermost electrons are five. Since the atoms want to have eight (the stable configuration), every two atoms pair, sharing three electron pairs (three covalent bonds) and they form the molecule N₂. The molecule itself is very stable, that means a lot of energy is needed to break those three bonds. But the interaction among the N₂ molecules is relatively weak, the so called van der Waals forces, and this way, N₂ is a gas at room temperature and it will not liquefy until -196 degree Celsius.

Please note that not always the change in one proton and electron produces such a dramatic change in the electronic properties. When you go to higher periods of the periodic table, among the metallic elements, variations are much smoother, again because of the very similar electronic structure of the adjacent elements that form very similar type of bond, known as metallic bond.

2. Well, this one if much tougher than the first one. The short answer is: We scientists do not have yet a really good explanation to this fact. All we know is that the attractive forces between mercury atoms are weaker that those in their neighbors. Since this question is often asked by many students, in a service similar to Scienceline offered by the scientists at Argonne National Laboratory I found a couple of interesting answers, please follow the link to:
click here

Hope to have helped you a little in getting a better understanding of the wonderful universe in which we live!