What defines an 'object' with regards to particle-wave duality? If any object, such as a ball, can exhibit wave behavior, I am confused about how such an object is defined. Does a ball itself have a wavelength? Does every single atom that composes it have a wavelength? Every single proton? Every single quark?
Can we then say that any combination of any particles in the ball, such as some atom on its left side and an electron on the right side, have their own wavelength as well?
Generally, and I know this is weirdly worded, what defines an object that can have a wavelength?
 A: In general, if you have $N$ particles, then the wavefunction is not a function on 3-dimensional space, so it is not a "wave" in the usual sense you might think of it from classical physics. The wavefunction is actually a wave in a $3N$-dimensional space (the $x, y, z$ coordinates of each of the $N$ particles). Therefore, to really visualize what the wavefunction looks like in its full complexity, you would need to think about a "wave" in an enormous space.
That's a lot of information to keep in your head at once. Instead, to keep things manageable, we try to find ways to project this information down to a simpler, but still meaningful, subspace. A very common way to do this is to only consider the center of mass degree of freedom. The center of mass in quantum mechanics is defined just as it is in classical physics
\begin{equation}
\vec{x}_{CM} = \frac{\sum_{i=1}^N m_i \vec{x}_i}{\sum_{i=1}^N m_i}
\end{equation}
where each particle has mass $m$ and position $\vec{x}_i$. Then we can project the full space down to just the three coordinates describing the center of mass. The wavefunction restricted to being a function of just the 3 center of mass coordinates, is what people normally have in mind when they talk about "the wavelength" of a macroscopic/composite object. (Incidentally, quantum superpositions have been measured for "macroscopic" objects such as buckyballs).
The Schrodinger equation also factorizes nicely in terms of the center of mass coordinate. For instance, if the potential is only a function of the separation of the particles, then we can write the full wavefunction as
\begin{equation}
\Psi(\vec{x}_1, \cdots, \vec{x}_N) = \psi(\vec{x}_{CM}) \Phi(\vec{x}_1 - \vec{x}_2, \cdots)
\end{equation}
where $\psi$ is a function only of the center of mass coordinate, and $\Phi$ is a function only of the relative coordinates. The center of mass wavefunction obeys the Schrodinger equation for a free particle
\begin{equation}
-\frac{\hbar^2}{2 M} \nabla_{CM}^2 \psi = E_{CM} \psi
\end{equation}
and the relative coordinate wavefunction obeys a Schrodinger equation with interactions between the particles encoded in the potential.
A: We don't know what nature really is, we can only make models of nature, some of which are more successful than others in explaining certain phenomena. What nature really is is a question for philosophy, but we can speak in terms of our theories, up to the point where they have been proven by experiment.

If any object, such as a ball, can exhibit wave behavior, I am confused about how such an object is defined. Does a ball itself have a wavelength? Does every single atom that composes it have a wavelength? Every single proton? Every single quark?

In quantum mechanics - disregarding general relativity - the answers are: yes, yes, yes, and yes.
In QM, each particle is a wave, so a system composed of many particles is also a big composed wave.
The catch is that the larger the object, the smaller its de Broglie wavelength (for non-relativistic speeds, $\lambda_{dB} = h / mv$), so it gets  very difficult to observe wave-like phenomena for anything larger than small molecules. Wikipedia cites an article where interference has been shown for a molecule with 2000 atoms, which has a de Broglie wavelength that is a thousand times smaller than the diameter of a hydrogen atom.
Now, if you consider general relativity, there's a limitation: if the object is large enough (about the size of a large bacterium), the wavelength becomes comparable to the object's Schwarzschild radius, and what happens in that case is a topic of debate.

Can we then say that any combination of any particles in the ball, such as some atom on its left side and an electron on the right side, have their own wavelength as well?

Yes, but note that, first, it can be impractical to perform an experiment to show wave-like properties of just a portion of a solid material, and, second, atoms in solid state may contribute to long range states of the material, like conduction bands in conductors, so these electrons have much larger wavelenghts.

Generally, and I know this is weirdly worded, what defines an object that can have a wavelength?

Its not everyday that you perceive particles as waves, that happens only in very specific scenarios, for instance, you are more likely to see wave-like behaviour of particles in setups where the characteristic lengths of the setup are comparable to de Broglie wavelength of the particle (see my answer here: What is exactly mean by wavelength in De Broglie equation?).
