I am convinced you would not have asked any of these questions had you gone through a fine texts, such as Modern Quantum Mechanics by Sakurai and Napolitano, for instance.
To avoid confusion with the $\hat \bullet $ notation, let us keep using that for unit vectors and use Capitals for operators and lower case for numerical variables, such as $r,\theta,\phi$.
As you may have learned, in the spherical coordinate representation, \begin{align} \mathbf L &= i \hbar \left(\frac{\hat{\boldsymbol{\theta}}}{\sin(\theta)} \frac{\partial}{\partial\phi} - \hat{\boldsymbol{\phi}} \frac{\partial}{\partial\theta}\right), \\ L^2 &= -\hbar^2 \left(\frac{1}{\sin(\theta)} \frac{\partial}{\partial\theta} \left(\sin(\theta) \frac{\partial}{\partial\theta}\right) + \frac{1}{\sin^2(\theta)}\frac{\partial^2}{\partial\phi^2}\right), \\ L_z &= -i \hbar \frac{\partial}{\partial\phi}~~~. \end{align}
Atomic wavefunctions are of the form $\langle x,y,z|n,l,m\rangle=R_{nl}(r) Y^m_l(\theta, \phi) $, and we normally use the normalization $\langle \hat r|l,m\rangle=Y^m_l(\theta,\phi)$, so that, using the standard "abuse" of notation for operator action on coordinate representations, $$ \langle \hat r|L^2|l,m\rangle= \hbar^2 l(l+1) Y^m_l(\theta,\phi) \\ \\= -\hbar^2 \left(\frac{1}{\sin(\theta)} \frac{\partial}{\partial\theta} \left(\sin(\theta) \frac{\partial}{\partial\theta}\right) + \frac{1}{\sin^2(\theta)}\frac{\partial^2}{\partial\phi^2}\right) Y^m_l(\theta,\phi),\\ \langle \hat r|L_z|l,m\rangle=\hbar m Y^m_l(\theta,\phi)= -i \hbar \frac{\partial}{\partial\phi} Y^m_l(\theta,\phi). $$
To address your questions then, the spherical operators you are talking about $R,\Theta,\Phi$ have eigenvalues $r,\theta,\phi$ but they need not be used explicitly. When acting on the wavefunctions we considered, they commute with each other and produce $\Theta Y^m_l(\theta,\phi)= \theta Y^m_l(\theta,\phi)$, $\Phi Y^m_l(\theta,\phi)= \phi Y^m_l(\theta,\phi)$ .
They most emphatically do not commute with $L^2$ and and (for $\Phi$) $L_z$, in general (except for $Y^0_0$), as evident above!
I wouldn't spend any time on the gradients $-i\hbar(\partial_r,\partial_\theta, \partial_\phi)$ which represent the respective canonical momentum operators, as you never need use them, in practice. The spherical harmonics $Y^m_l$ obey well known orthogonality relations, where you must recall the angular measure involves a factor of $\sin\theta$.