As regards your first question, the space of tempered distributions is indeed an example of a special class of locally convex spaces which, while not metrisable, have very nice properties. These are the Silva spaces which were investigated by the portuguese mathematician Sebastiao e Silva---they are, by definition, inductive limits of sequences of Banach spaces with compact interconnecting mappings---in your case, these are even nuclear.
The original articles are rather inaccessible but Koethe's monograph on topological vector spaces has a chapter on these spaces. Since everything in sight in your application is nuclear, the operator spaces you are interested in can be represented as tensor products (in any of the standard tensor product topologies---in this case they all coincide). These facts were investigated in considerable detail by some of the great masters of functional analysis, for example, Koethe, Schwartz and Grothendieck and there is a wealth of material on this topic in their works---for example, Koethe's treatise mentioned above, Schwartz'
sequel to his classical "Th\'eorie des Distributions" (in which he considers vector-valued distributions---this is easily available online) and Grothendick's thesis. One can find more accesssible representations in the recent secondary literature on locally convex spaces.
There are two different views about the semiclassical limit in quantum mechanics, the first is based on a somewhat shaky ground due to the fact that the existence of the Feynman integral is not proved yet. On the other side, Wiener integral, its imaginary time counterpart does exist and one could pretend to work things out from this and then move to the Feynman integral. The other approach relies on substantial mathematical theorems due to Elliott Lieb and Barry Simon in the '70 and is essentially valid for many-body physics. These latter results make the limit $\hbar\rightarrow 0$ and $N\rightarrow\infty$ equivalent while the former is not really a physical limit due to the fact that Planck constant is never zero.
Starting from Feynman path integral, the standard formulation applies to a mechanial problem described from a Lagrangian $L$, normally $L=\frac{\dot x^2}{2}-V(x)$ but one can extend this to more general cases, and then the postulate is that, given a path $x(t)$, this must contribute to the full quantum mechanical amplitude of a particle going from the point $x_a$ to $x_b$ with a term $e^{\frac{i}{\hbar}S}$ being $S=\int_{t_a}^{t_b}dtL(\dot x,x,t)$ the action. All the possible paths contribute and so, the full amplitude will be given by the formal writing
$$
A(x_a,x_b)\sim\int[dx(t)]e^{\frac{i}{\hbar}\int_{t_a}^{t_b}dtL(\dot x,x,t)}.
$$
Be warned that this integral is not proved to exist yet, but the Wiener counterpart, that can be obtained changing $t\rightarrow it$, exists and describes Brownian motion. Now, if you take the formal limit $\hbar\rightarrow 0$ to this integral you will immediately recognize the conditions to apply the stationary phase method to it. This implies that the functional must have an extremum and this can be obtained by pretending that
$$
\delta S=\delta \int_{t_a}^{t_b}dtL(\dot x,x,t)=0
$$
that is, the paths that give the greatest contribution are the classical ones and one recover the classical limit as a variational principle as learned from standard textbooks.
While this is a quite common approach, to extend what really happens to a macroscopic system that we can see to respect all the laws of classical mechanics, we have to turn our attention to the limit of a large number of particles $N\rightarrow\infty$. In this case one has more rigorous results. These are due to Lieb and Simon as already said above. They published two papers about
Lieb E. H. and Simon B. 1973 Phys. Rev. Lett. 31, 681.
Lieb E. H. and Simon B. 1977 Adv. in Math. 23, 22.
In the first paper, their theorem 4 states
Theorem: For $\lambda < Z$, let $E_N^0$ and $\rho_N^0(x)$ denote the ground-state energy and one-electron distribution function for N spin-$\frac{1}{2}$ electrons obeying the Pauli principle and interacting with $k$ nuclei as described above. Then (a) $N^{-\frac{7}{3}}E_N^0\rightarrow E_1$, as $N\rightarrow\infty$; (b) $N^{-2}\rho_N^0(N^{-\frac{1}{3}}x)\rightarrow\rho_1(x)$ as $N\rightarrow\infty$, where convergence in (b) means that for any domain $D\subset R^3$, the expected fraction of electrons in $N^{-\frac{1}{3}}D$ approaches $\int_D\rho_1 (x)d^3x$.
Where $\rho_1(x)$ and $E_1$ refer to the Thomas-Fermi distribution and the corresponding energy. This theorem states that the limit $N\rightarrow\infty$ for a quantum system, under some mild conditions, is the Thomas-Fermi distribution. A system with this distribution is a classical system. The fact that a system with a Thomas-Fermi distribution is a classical one can be seen through the following two references:
W. Thirring(Ed.), The Stability of Matter: From Atoms to Stars - Selecta of E. Lieb, Springer-Verlag (1997).
L. Hörmander, Comm. Pure. Appl. Math. 32, 359 (1979).
The second paper just gives the mathematical support to derive Thomas-Fermi approximation as the leading order of a classical expansion for $\hbar\rightarrow 0$ that I will not present here.
Best Answer
In addition to what has already been said I would like to add some more comments. I completely understand your suspicion that the passage from unbounded operators to bounded ones is at least tricky. For the canonical commutation relations of position and momentum operators this can be solved in a reasonable and also physically acceptable way by passing to the Weyl algebra (OK; there are zillions of Weyl algebras around in math, but I'm refering here to the $C^*$-algebra generated by the exponentials of $Q$'s and $P$'s subject to the heuristic commutation relations arising from $[Q, P] = \mathrm{i} \hbar$).
However, there are other situations in physics where this is much more complicated: when one tries to quantize a classical mechanical system which has a more complicated phase space than just $\mathbb{R}^{2n}$ then one can not hope to get some easy commutation relations which allow for a Weyl algebra like construction. To be more specific, for general symplectic of Poisson manifolds the beast quantization scheme one can get in this generality is probably formal deformation quantization. Here the classical observable algebra (a Poisson algebra) is deformed into a noncommutative algebra in a $\hbar$-dependent way such that the new product, the so-called star product, depends on $\hbar$ in such a way that for $\hbar = 0$ one recovers the classical mutliplication and in first order of $\hbar$ one gets the Poisson bracket in the commutator.
Now two difficulties arise: the most severe one is that in this generality one only can hope for formal power series in $\hbar$. Thus one has even changed the underlying ring of scalars from $\mathbb{C}$ to $\mathbb{C}[[\hbar]]$. So there is of course no notion of a $C^\ast$-algebra whatsoever on this ring. Nevertheless, there is a good notion of states in the sense of positive functionals already at this stage. Second, even if one succeeds to find a convergent subalgebra one does usually not end up with a $C^\ast$-algebra on the nose. Worse: in most of the explicit exampes one knows (and there are not really many of them...) the commutation relations one obtains are very complicated. In particular, it is not clear at all how one can affiliate a $C^\ast$-algebra to them. Moreover, it is not clear which of the previous states survive this condition of convergence and yield reasonable representations by the GNS construction.
It takes quite some effort to first represent the observables by typically very unbounded operators in a reasonable way and then show that they give rise to some self-adjoint operators still obying the relevant commutation relations. This is far from being obvious. To get a flavour for the difficulties it is quite illustrative to take a look at the book of Klimyk and Schmüdgen on Quantum Groups adn their Representations. Note that in these examples one still has a lot of structure around which helps to understand the analysis.
However, physics usually requires still much more general situations. Most important here are systems with gauge degrees of freedoms where one has to pass to a reduced phase space. Even if one starts with a geometrically nice phase space the reduced one can be horribly complicated. This problem is present in any contemporary QFT really relevant to physics :(
For other quantization schemes things are similar, even though I'm not quite the expert to say something more substantial :)
So one may ask the question why one should actually insist on $C^\ast$-algebras and this strong analytic background. There is indeed a physical reason and this is that quantum physics predicts not only expectation values of observables in given states (here the notion of a ${}^\ast$-algebra and a positive functional is sufficient) but also the possible outcomes of a measurement: they are given by particular numbers called the physical spectrum of the observable. To get a good description with predictive power the (as far as I know) only way to achieve this is to say that the physical spectrum is givebn by the mathematical spectrum of a self-adjoint operator in a Hilbert space. If one is here at this point, then the passage from a (unbounded) self-adjoint operator to a $C^\ast$-algebra is comparably easy: one has the spectral projections and takes e.g. the von Neumann algebra generated by them...
So the point I would like to make is that it is very desirable from a physical point of view to have the strong analytic framework for observables as either self-adjoint operators or Hermitian elements in a $C^\ast$-algebra. But the quantum theory of many nontrivial systems requires a long long and nontrivial way before one ends up in this nice heavenly situation.