First of all, in any atom, the number of protons in the nucleus must equal the number of electrons out side of the nucleus. And the behavior of the electrons must obey a series of rules under quantum mechanics. There is one particular rule of quantum mechanics called "Pauli's exclusion principle", this means that any two electrons cannot have exactly the same behavior in an atom. This directly leads to the fact that adding one electron (also one proton) to the atom can potentially drastically change the property of the atom, because this one extra electron has to behave totally differently from all the other electrons, due to the Pauli's principle.

The protons all carry positive charge, so it is hard to imagine how they can all stick together instead of pushing each other away in a nucleus. Indeed, the Coulomb force dictates that the same charges repel each other. However, the protons also have what is called the "strong force" between each other. And it is the strong force that holds all the protons together (actually the strong force holds together all the protons with many neutrons in a nucleus). The size of a nucleus is extremely small (roughly 1/100000 of an atom), at that length scale, the strong force will dominate the Coulomb interaction. But strong force is a very short-range force, beyond the scale of the nucleus, strong force can be ignored. While the Coulomb force is a long range force. So in our daily life the strong force is negligible, we only need to worry about the Coulomb force.

By the way, there are four so called fundamental forces in our universe: gravity, electromagnetic force (more general version of the Coulomb force), strong force, and weak force. The first two (gravity and Coulomb forces) are long range forces, this is why they dominate most of the phenomena in our life; the strong and weak forces are very short range forces, they will not play a role unless we study very tiny objects like nucleus.

Changing the number of protons in an atom is important to the properties because it changes the number of electrons in the stable configuration of the atom. This is because the properties of an atom are determined by the way it interacts with other atoms. The interactions occur through chemical bonds, which are intimately linked to the number of electrons. Bonds form when doing so is energetically favorable . Adding or removing electrons will change the nature of the bonds which form (or even make formation of bonds unfavorable ), thereby changing the properties. An important point is that it is (primarily) the electrons in the outer shell which are involved in bonding. Combining this with the previous information, one might expect that atoms with the same number of electrons in their outer shell ("valence electrons") to have similar properties. In fact, this is the case - columns of the periodic table are called "groups" because the elements within them tend to have very similar properties, and this stems from their having the same number of valence electrons and thus forming similar bonds.

There are differences between elements within a group though, and these are also typically related to the number of electrons. Here it is the total number of electrons though, not just the number in the outer shell. For example, boiling points tend to increase with atomic number (i.e., with number of protons, moving down a column). This is because the attractive forces holding the atoms together ( Van der Waals forces , comprising dipole-dipole and London dispersion forces) are related to the average position of the electrons. With a larger number of electrons, larger dipoles can be created, thus causing stronger attractions and increasing the energy needed to pull them apart.

The second question, about electrons staying near the nucleus is a bit tricky. The main answer is that electrons are kept near the nucleus due to electromagnetic interactions - the negative charge of the electron is attracted to the positive charge of the proton. However, there are some other considerations/complications. The common picture of atoms (a holdover from the early days of atomic physics) is one in which electrons orbit the nucleus , much like planets around a sun. This model has plenty of issues , but the end result is that it is better to consider electrons to be a sort of distribution around the nucleus rather than a particle with an identifiable proton.

Opposite electric charges exert a force that draws them together. Protons carry one unit of positive charge, while electrons carry one unit of negative charge. Thus an atom with more protons than electrons will have a net positive charge and attract electrons until the protons and electrons are equal in number and the atom has a total charge of zero.

A force called the strong force holds neutrons and protons together in the nucleus. Electrons are held near the nucleus due to the attraction between the positively charged protons and negatively charged electrons. When a proton is added to an atom, it makes the nucleus more massive — a proton is nearly 10000x as massive as an electron. This causes the element to weight more.

The increased positive charge also causes the electrons to be more attracted to the nucleus. If an electron is added, the electron cloud which surrounds the nucleus becomes bigger, and the atom has more electrons with which to interact with its surroundings. This means it can bond to other atoms, or distort the electron clouds of nearby atoms. Physical properties like boiling point, freezing point, and even the color of the element all have to do with how much the nucleus weighs, and how the electrons are interacting with the environment. The exact interactions are generally very complex, but you can imagine that when you have a lot of atoms - most every day examples of chemical systems contain on the order of 10₍₂₃₎ atoms - one more electron on each atom can make a big difference!