Do subatomic particles in atoms change shape? Do protons, neutrons and electrons change shape?
If they don't:
What shape are they individually? Are they irregularly shaped? Are they 'regularly shaped'? Spherical (like in textbooks, diagrams)? Cube-shaped? Prism-shaped? Are all protons/neutrons/electrons the same shape? Do some protons/neutrons/electrons have different shapes than other protons/neutrons/electrons respectively?  Do protons and neutrons have the same shape when in atomic nuclei?
If they do: How do they change shape whilst maintaining constant mass? Is their volume constant? Can we predict their shapes or is it random? How do the subatomic particles change shape to not collide with each other? Do protons and neutrons change shape in the nuclei? Do electrons change shape in electricity when they drift through a circuit?
Either way, can we predict the amount of empty space taken up inside any given atom (the borders of the atom being the outline of its highest energy electron's orbital/subshell)? Can we find the volume of the subatomic particles themselves in order to find the amount of empty space? Do we know the area of space around a nucleus that the electrons physically cannot go outside of (the electrons themselves still have a small probability of falling outside their assigned orbitals) in order to find where an electron could be at any given moment? When alpha and beta radiation occur, how do the electrons in the atom take a position in such a way that the alpha/beta particles being ejected don't collide with the electrons orbiting the nucleus (do they change shape)?
 A: At the subatomic level, the concept of "shape" doesn't really apply. Things aren't solid objects which you can see and which have discernible properties. They are tiny little packets of energy whizzing around and which you can't really observe fully in the first place. Even concepts like mass have slightly different definitions at that scale.
To answer your question - they don't change shape because they don't really have a shape in the first place.
A: 
Do protons, neutrons and electrons change shape ?

In the Standard Model of particle physics, elementary particles such as the electron are modelled as point particles. Protons and neutrons are not elementary particles, but instead each consist of three quarks. The quarks are elementary particles, and are again modelled as point particles. So in the Standard Model, protons, neutrons and electrons do not have shapes.
We suspect this is not the whole story, and there are theoretical extensions to the Standard Model such as string theory in which the elementary particles do have shapes. However, they are not simple three-dimensional shapes such as spheres or cubes. In string theory the elementary particles are one-dimensional strings which are wrapped around themselves in complicated ways in a space of 10 or 11 dimensions.

Do we know the area of space around a nucleus that the electrons physically cannot go outside of ?

There is no simple answer to this question. Due to the stochastic nature of quantum physics, the electron cloud around an atom does not have a sharp edge. Even if we ask "how large is the region within which we are 99% certain to find the electrons of an atom", the answer depends on the element concerned and whether the electrons are in an excited state. In addition, if the atom is part of a molecule, then its electrons may be spread out into a larger molecular orbital.
