First, there is a great survey of locally convex topological spaces in section 424 of the Encyclopedic Dictionary of Mathematics. (The EDM, if you have not seen it, is a fabulous reference for all kinds of information. Even though we all have Wikipedia, the EDM is still great.) At the end it has a chart of many of these properties of topological vector spaces, indicating dependencies, although not all of them. This chart helped me a lot with your question. On that note, I am not a functional analyst, but I can play one on MO. Maybe a serious functional analyst can give you a better answer. (Or a worse answer? I wonder if to some analysts, everything is the machine.)
Every important topological vector space that I have seen in mathematics (of those that are over $\mathbb{R}$ or $\mathbb{C}$) is at least a Banach space, or an important generalization known as a Frechet space. We all learn what a Banach space is; a Frechet space is the same thing except with a family of seminorms instead of one norm. Many of the properties that you list, for instance metrizable and bornological, hold for all Frechet spaces. In emphasizing Banach and Frechet spaces, the completeness property is implicitly important. Since a normed linear space is a metric space, you might as well take its completion, which makes it Banach. A Frechet space is a generalization of a metric space known as a uniform space and you might as well do the same thing. Also the discussion is not complete without mentioning Hilbert spaces. You can think of a Hilbert space either as a construction of a type of Banach space, or a Banach space that satisfies the parallelogram law; of course it's important.
Of the properties that do not hold for all Frechet spaces, I can think of four that actually matter: reflexive, nuclear, separable, and unconditional. In addition, a Schwarz space isn't really a space with a property but a specific (and useful) construction of a Frechet space.
A discussion of the properties that I think are worth knowing, and why:
reflexive. This means a space whose dual is also its pre-dual. If a Banach space has a pre-dual, then its unit ball is compact in the weak-* topology by the Banach-Alaoglu theorem. In particular, the set of Borel probability measures on a compact space is compact. This is important in geometry, for sure. Famously, Hilbert spaces are reflexive. Note also that there is a second important topology, the weak-* topology when a pre-dual exists, which you'd also call the weak topology in the reflexive case. (I am not sure what good the weak topology is when they are different.)
separable. As in topology, has a countable dense subset. How much do you use manifolds that do not have a countable dense subset? Inseparable topological vectors are generally not that useful either, with the major exception of the dual of a non-reflexive, separable Banach space. For instance $B(H)$, the bounded operators on a Hilbert space, is inseparable, but it is the dual of the Banach space of trace-class operators $B_1(H)$, which is separable.
unconditional. It is nice for a Banach space to have a basis, and the reasonable kind is a topological basis, a.k.a. a Schauder basis. The structure does not resemble familiar linear algebra nearly as much if linear combinations are only conditionally convergent. An unconditional basis is an unordered topological basis, and an unconditional space is a Banach space that has one. There is a wonderful structure theorem that says that, up to a constant factor that can be sent to 1, the norm in an unconditional space is a convex function of the norms of the basis coordinates. All unconditional Banach spaces resemble $\ell^p$ in this sense. Note also that there is a non-commutative moral generalization for operators, namely that the norm be spectral, or invariant under the available unitary group.
nuclear. Many of the favorable properties of the smooth functions on a compact manifold (a Schwartz space) come from or are related to the fact that it is a nuclear Frechet space. For instance, in defining it as a Frechet space (using norms on the derivatives), you notice that the precise norms don't matter much. This doesn't necessarily mean that you should learn the theory of nuclear spaces (since even most analysts don't). But in drawing a line between constructions and properties, my impression is that the main favorable properties of a Schwartz space are that it is Frechet, reflexive, and nuclear. On the other hand, infinite-dimensional Banach spaces are never nuclear.