Conceptual and practical reasons and consequences of inverting weak equivalences Although dealing with this in one or other form for many years, to my shame this question only struck me now.
One of the most radical differences between categories of "algebraic" and "topological" kind is absence/presence of nontrivial bimorphisms, i. e. non-invertible morphisms which are both mono and epi. Such guys exist in e. g. distributive lattices or rings, posets or topological spaces; they are absent in e. g. groups or compact Hausdorff spaces. Important case is that of quasitoposes: presence of nontrivial bimorphism is what distinguishes them from toposes.
Now, in homotopical algebra one inverts weak equivalences, and "morally" it very much looks like inverting bimorphisms, although there is no rigorous reason at all that weak equivalences be related to bimorphisms at all, and this is the first thing I don't quite understand:

Is there any connection between inverting weak equivalences and inverting bimorphisms?
Does one obtain any recognizable category by inverting bimorphisms of, say, posets or commutative rings?

Independently whether weak equivalences are "like bimorphisms" or not - ok but what are they anyway? That is, in what way do we improve a, say, model category when we invert weak equivalences? Another example where my ignorance is especially apparent is the case of homotopy categories of complexes versus derived categories. In the former we identify homotopic maps, in the latter - additionally invert quasiisomorphisms. In both cases we obtain triangulated categories. I have no idea how the latter (i. e. the derived category) is a "better" category than the former (i. e. the homotopy category). They both are triangulated, which basically means that all "nice" category-theoretic properties related to limits, etc. are destroyed anyway. So what I don't understand here is this -

Are there some abstract category-theoretic properties which are improved by passing from a homotopy category of chain complexes to the derived category?

 A: I'm sorry, but I think this question is too broad and vague. There is a tremendous amount of literature on localization and on the derived category, and it's impossible to summarize. The book of Gabriel-Zisman, work of Dwyer-Kan in the 80s, and work of Barwick-Kan on relative categories tells you when you can localize and why you would want to. With the machinery of relative categories, you are certainly free to invert the bimorphisms (and a bit more, including their closure under the two out of three property), and your reasons for doing so (that they are "bad things" that you want to invert) is perfectly acceptable in homotopy theory. My point is that localization is a fundamental thing in homotopy theory, in some sense it is the entire point. I recommend reading the introduction to this paper:
https://www.maths.ed.ac.uk/~cbarwick/papers/relcats-mod.pdf
As for the difference between $D(R)$ and $K(R)$, this is detailed very nicely in Weibel's book "An introduction to homological algebra," as well as many other places. Both categories are useful in homological algebra and representation theory, but $D(R)$ is more useful because it captures the "true" homological algebra rather than a naive one (because, in homological algebra, it's more common to work up to quasi-isomorphism). It's used to compute derived functors a la Grothendieck, and for zillions of other purposes. Just googling "why is the derived category better?" or "what is the derived category good for" you find tons of books, articles, lecture notes, blog posts, mathoverflow answers, etc. 
A few other remarks on your question. I would not say we "improve" a model category when we invert the weak equivalences. Rather, the fundamental thing we want to do as homotopy theorists is invert the weak equivalences, and the model category gives us control over the new mapping spaces in the homotopy category, as well as ways to compute homotopy colimits and other homotopical objects of interest. 
When you write "they both are triangulated, which basically means that all "nice" category-theoretic properties related to limits, etc. are destroyed anyway", I feel like you are missing the point of being triangulated. It certainly doesn't "mean" that we are destroying nice category-theoretic properties, i.e. we didn't invent it to make our lives harder. It's a critical object of study that arises naturally from the fundamental idea of localization, and we homotopy theorists put a lot of effort into making it easier to work with, e.g. via model structures. 
