I am not the perfect person to answer this, as am still an active student of Statistical Mechanics, but I believe I can provide some thoughts. Also, I made an attempt to break it down in terms of your questions, but for me they are all related.
Question 1
The relationships between Statistical Mechanics (or the fundamental ideas within) and Classical Thermodynamics can be established multiple ways. They all, however, share the same basis in the thermodynamic limit (that the particle density is constant as $N\rightarrow \infty, V\rightarrow \infty$).
Ontologically, I would answer this question by examining the roles that the microcanonical and canonical ensembles play as thought experiments.
The microcanonical ensemble ($NVE$ ensemble), informs us of the relationship between thermodynamic temperature, $\beta$, and normal temperature, $T$, namely, $\beta \propto 1/T$. It also provides us with the early definition of entropy which is etched into Boltzmann's tombstone, $S=k_B ln \Omega$. This can be established from the single axiom that the number of microstates at equilibrium is maximized,
$$\frac{\partial \Omega}{\partial E}(E=\bar E)=0$$
While this tells us the temperature is constant for two systems in thermal contact in an $NVE$ ensemble, it's only natural to ask what would happen if the energy was unbounded and the temperature was held constant. This is the canonical ensemble ($NVT$ ensemble) and it is physically imagined as a system in contact with a heat bath. Through its various derivations, a probability equation is provided and appears to be something close to the fundamental thermodynamic equation when taking its logarithm:
$$P_r = P_0 \exp(-\beta E_r)= \exp(\beta F - \beta E_r) \tag{1}$$
The reason it doesn't align with the fundamental thermodynamic equation is because the probability is provided as a function of a particular energy state, $E_r$, and not the average energy. There are, again, a multitude of ways of reasoning that the ensemble average, the time average, and the expectation of experiment are all closely related, and we can observe this through manipulations of the canonical ensemble. In any case, we learn we can take the ensemble average of $f$, as,
$$\langle f \rangle = \sum_r f P_r$$
There a couple of ways of "expanding" the notion of a canonical ensemble to reach the desired thermodynamic quantities. One, take the Legendre transform of $\{S/k_B,\bar E \} \rightarrow \{\beta F,\beta\}$. By calculating the partial derivatives of either Legendre Transform variable you can confirm its validity. Second, you can use (and validate, by induction) Gibbs' entropy formulation, $S=-k_B\sum_r P_r \ln P_r$. Either, combined with (1), give us an analogy between the thermodynamic equation and our theoretical quantities:
$$d\bar E=TdS-PdV+\mu dN$$
$$F=\bar E - TS$$
$$dF=d\bar E-TdS-SdT=-PdV+\mu dN - SdT$$
There are many extensions of these results, from physical quantities of the ideal gas and beyond.
The importance of this first question is the whole reason the field exists. I try to remind myself that while Boltzmann and Gibbs (and others) were reasoning about S.M., there was not even a scientific consensus on the idea of atoms and molecules as fundamental constituents of matter.
Question 3
I mentioned a bit about the $NVE$ ensemble above. The only addition I have is that the same process that can be used to determine the relationship between $\beta$ and $T$ can be used for the other variables ($V$ and $N$), demonstrating that $P$ and $\mu$ are constants under equilibrium. This allows us to establish all the normal partial derivatives associated with the $S$, $E$, $V$, etc. for two-systems in contact.
For the $NVT$ ensemble, there are a couple of interesting ways to theoretically discuss the idea. One, through Liouville's Theorem, and two through combinatorics (later).
Liouville's Theorem establishes a few interesting results: first, that the phase space acts as an incompressible fluid (Gibbs was actually the first to reason about this in the context of S.M.), second, the requirement of statistical equilibrium:
$$\frac{\partial P_r}{\partial t}=0 \tag{2}$$
This "axiom" will provide us with (1). More on this in Q4 and Q5.
Things I am less knowledgable about:
Question 2
I won't say much here, other than these variables, extensive and intensive, can be examined in detail by expanding on notions mentioned in question 1. Physical quantities such as specific heat capacities, pressure, and temperature can be related to S.M. quantities such as entropy, and the various free energies.
Question 4
The connection to the kinetic theory of gases should be fairly obvious, as S.M. predicts those ideas clearly (ideal gas law, Van der Waals model, average kinetic energy of gases, Maxwell's distribution, etc.).
To me, the connection with hydrodynamics is a much subtler one. Since Liouville's Theorem implies that there is a conserved current of probability-phase-space (for equilibrium or otherwise), we can draw many parallels to fluid flow equations. In fact, there is a continuity equation for S.M., just as there is for fluid dynamics as well as electromagnetism.
Question 5
Establishing the $NVT$ ensemble from a combinatorics viewpoint provides us with the exact same result as Liouville's Theorem, but with the new idea of energy levels. In my opinion it more strongly motivates the idea of probability, defining $P_r$ as the fraction of the number of systems with energy, $E_r$, to the total number of systems. This reasoning about the canonical ensemble (we learn) is more general than that of just thermodynamic systems, and extends to systems containing other, non-Newtonian particles. Probably most evidently demonstrated in the resolution of the ultraviolet catastrophe, which applied $E_r=n(r)h\nu$ to the canonical ensemble to obtain the correct result for the spectral irradiance from a black body.
Question 6
In relation to Liouville's Theorem and the Hamiltonian we observe a conserved "probability current", denoted by the n-tuple $(P_r \dot{q_i}, P_r \dot p_i)$. The results of equilibrium S.M. as discussed above truly do fall out from this statement alone, and in the non-equilibrium case we have the task of dealing with (2) no longer being true.
