I got confused while reading about dielectrics, so basically my question is:
(a) what's the difference between a (homogenous and isotropic) dielectric and (linear) dielectric? Does the first imply the second? I know that linear dielectrics obey $P=\epsilon_0(\chi_e)E$ but does this imply that the dielectric is homogenous and isotropic? If not why not?
A homogeneous dielectric does not change its value (significantly) at different points across its area or volume. A possible example of the opposite (inhomogeneous dielectric) would be for example suspended material in a liquid (e.g. small particles in a suspension, think maybe gold nanoparticles in a toluene solution).
The description for a time-dependent dielectric function would then be:
$\epsilon_e(\vec r,\vec r',t) = \epsilon_e(\vec r=0,\vec r'=0,t)$
Where $\epsilon_e = \epsilon_0 \chi_e$ to follow your convention.
An isotropic material has the same dielectric function along all directions. As a counterexample, anisotropic material may have at least one direction which is not the same as the other directions, especially when interacting with light.
$\epsilon_e(x,t) \neq \epsilon_e(y,t)$
This is very common for electro-optic materials and so-called birefringent materials, where you have the "fast and slow" axes, the "ordinary and extraordinary" refractive index, and many other interesting descriptions. Lithium niobate is an inorganic material which is very commonly referred to in the literature due to this birefringence and the possibility to control it, for example, for modulation.
A linear dielectric refers to the order of the dielectric function and roughly it has as counter examples the nonlinear dielectric function which are higher-order dielectric functions which strongly depend on the power of the electromagnetic wave interacting with the material (e.g. high power lasers or pulsed lasers).
A material can be homogeneous and isotropic but nonlinear. Additionally, anisotropy does not imply inhomegeneity or nonlinearity. Lithium niobate, for example, is a material that is both nonlinear and anisotropic but it is homogeneous (if properly manufactured).
P=ϵ0(χe)E but does this imply that the dielectric is homogenous and isotropic?
From that equation, not necessarily. You would need to describe the dielectric function for its spatial and temporal properties to imply such a thing:
$P = \epsilon_e(\vec r,\vec r',t)E + \epsilon_e^2(\vec r,\vec r',t)EE$
Here I generally describe a material that can be anisotropic, inhomogeneous and nonlinear. By reducing the assumptions (no spatial dependency of the dielectric function, limited spatial dependency and only operating at low laser powers) you can reduce the equation to your initial Ansatz.
P=ϵ0(χe)E
But then again that would be more or less how you get to that part of the equation.
