Quantum master equation and off-diagonal terms

I have a couple of related questions

1. What is exactly the difference between the quantum master equation and the regular master equation? My understanding is that the normal master equation is used to find a "vector" of state probabilities (like in a regular Markov chain), whereas in the quantum master equation one finds the density matrix. Is this correct? If this is the case, how does the "transition matrix" look, in the quantum case?
2. I'm also a bit confused about the off-diagonal elements in a density matrix. As the density matrix is self-adjoint it can be diagonalizable in some orthonormal basis. So why do we speak about off-diagonal elements? Is it because the density matrix $\rho$ can be time dependent (and orthonormal basis stop being so as time evolves $\Rightarrow$ off-diagonal elements appear)?

The off-diagonal terms appear when you analyze the measurement problem. Say you have a system S with some observable $\hat{S}$ such that it haves eigenvalues $s_i$ and you wish to measure the state. In order to do that you consider an apparatus $\hat{A}$ wich is initially in a pointer state $| a_0 \rangle$ so the initial state of your system is given by the tensor product:

$|\Psi \rangle = |s \rangle \bigotimes |a_0 \rangle$

The act of measurement is the interaction between the apparatus and your system so schematically your Hamiltonian will be:

$H = \hbar \Omega \sigma_z \bigotimes z$

But in general that interaction will form an EPR like state between the system and the apparatus:

$|\Psi \rangle = |s \rangle \bigotimes |a_0 \rangle \mapsto \sum c_i |s_i \rangle |a_i \rangle$

So now your density matrix has off diagonal terms wich in general you don't see. So how do we kill them? Via decoherence from the environment