First you say
It's easy to visualise and comprehend the excited states of electrons, because they exist on discrete energy levels that orbit the nucleus
By way of preparation, I'll note that in introductory course work you never attempt to handle the multi-electron atom in detail. The reason is the complexity of the problem: the inter-electron effects (screening and so on) mean that it is not simple to describe the levels of a non-hydrogen-like atom. The complex spectra of higher Z atoms attest to this.
Later you say
[nuclei] don't exist on energy levels that they can transfer between
but the best models of the nucleus that we have (shell models) do have nucleons occupying discrete orbital states in the combined field of the all the other nucleons (and the mesons that act as the carriers of the "long-range" effective strong force).
This problem is still harder than that of the non-hydrogen-like atoms because there is no heavy, highly-charged nucleus to set the basic landscape on which the players dance, but it is computationally tractable in some cases.
See my answer to "What is an intuitive picture of the motion of nucleons?" for some experimental data exhibiting (in energy space) the shell structure of the protons in the carbon nucleus. In that image you will, however, notice the very large degree of overlap between the s- and p-shell distributions. That is different than what you see in atomic orbitals because the size of the nucleons is comparable to the range of the nuclear strong force.
You are not correct in your latter part of the analysis; the chemical properties (which is mostly what matters in ordinary matter) almost only depend on the electron shell, and in particular the outermost electrons (called the valence electrons).
So more protons mean more electrons and a different electron shell, meaning different chemical properties.
Why there is such a diversity of properties just by changing around the electron shell, is one of the wonders of chemistry! Due to quantum mechanics, the electrons don't simply spin around the nucleus like planets around the sun, but arrange themselves in particular, complicated patterns. By having different patterns, you can achieve a lot of different atom<->atom binding geometries, at a lot of different energies. This is what gives the diversity of chemical properties of matter (see the periodic table).
You can add or remove electrons to an atom to make the electron shells look more like the shells of another atom (with a different number of protons), but then the atom as a whole is then no longer electrically neutral, and due to the strength of the electromagnetic force, the resulting ion does not imitate the other atom type very well (I'm not a chemist - I'm sure there are properties that indeed could become similar).
Many physical properties are also mostly due to the electron shells, like photon interactions including color. Mass obviously is almost only due to the nucleus though, and I should add that in many chemical processes the mass of the atoms are important for the dynamics of processes, even if it isn't directly related to the chemical bindings.
This was just a small introduction to chemistry and nuclear physics ;)
Best Answer
Atoms are formed by a nucleolus with Z protons and an equivalent Z number of electrons. The nuclear mass (A) equals the mass of its protons plus the mass of its neutrons minus the mass lost as energy when binding all these protons and neutrons in the atom nucleus. When an atom loses $\rm n$ number of electrons, it becomes an ion with a ${\rm +n}e$ charge or ${\rm -n}e$ charge otherwise, with $e$ being the fundamental charge.
There are two ways protons can be bound close to each other:
The first form is for example the case that happen when a proton with an electron (Hydrogen atom) bounds to an equal partner by way of covalent forces and creates the hydrogen molecule ($\rm H_2$). In this setup, these protons could not be bound together without its electrons since electron sharing is causing the bound. Instead of hydrogen atom this example could be made using two deuterium atoms forming a deuterium molecule with 2 protons, 2 neutrons and 2 electrons.
The second form is that protons could be bound together by means of strong nuclear force. This is one of the four fundamental forces and is of a different nature than the covalent bound used before. Therefore, the two protons of the Helium ($\rm He$) atom are not bound by the same forces as the two protons in the deuterium molecule, even though both structures have 2 protons, 2 neutrons and 2 electrons. This is the reason why they can exist even without electrons in the form of an helium ion ($\rm He^{2+}$).
It’s stability is accounted by the immense energy that is lost when the four nucleons are bound together. Mathematically speaking,
$${\rm ΔE=(2m_p+2m_n-m_{He^{+2}})}c^2$$
Where,
$\rm ΔE$ is the alpha particle’s ($\rm He^{+2}$) binding energy, $\rm m_p$ is the proton mass, $\rm m_n$ is the neutron mass, $\rm m_{He^{+2}}$ is the alpha particle mass and $c$ is the speed of light.
Thus, one needs to supply all that energy back to the $\rm He^{2+}$ nucleus in order to dismantle it.
PS: The particle $\rm He^{2+}$ when emitted by some radioactive materials is known as alpha radiation or alpha particle.