There are two places in the curriculum where point-set topology is taught. The first is a course in "general topology". Here the students have (hopefully) seen the basic topology of metric spaces (eg in Rudin's small book). Books intended for this audience (such as Munkres's book, which seems to be the gold standard) often omit nets and filters. I don't know of any written explanation from eg Munkres why he made this choice, but I can speculate. The typical student here is greatly inclined to think of topological concepts in terms of sequences. Teaching them notions of generalized convergence would be misleading. Given their lack of experience, they would probably think of eg nets as "just generalized sequences", not appreciate the subtleties of things like subnets, and in the end not appreciate the strange things that can happen in arbitrary topological spaces. Moreover, they would probably not learn to think of things like continuity in terms of open sets, which is much more elegant and conceptual and also quite important in applications (eg in algebraic geometry) where you are dealing with spaces that are very much not metric spaces.
The other place where point-set topology is taught is during functional analysis courses. Here certainly many standard books (like Reed-Simon) use things like nets, and this makes sense since the students are typically more mathematically sophisticated when they take these courses.
For each of the functions you mention, the answer to "why it's useful" will be somewhat different. In general, if a function appears in many unrelated contexts, that's a sure sign that it's useful. Think of other mathematical structures that appear in many unrelated contexts: the natural numbers, the real numbers, the complex numbers, polynomials, trigonometric functions, etc.
My understanding is that in the olden days, the second half a complex analysis course - after the theory has been developed - was a detailed study of special functions. (At least this may have been the case in courses for physicists.) That doesn't seem to be so common anymore.
Some remarks about specific functions that you mention:
Gamma function: Just as it turned out to be very convenient to allow the $x$ in $a^x$ to take non-natural-number values, it is convenient to allow the $n$ in $n!$ to do so. To give but one small example of an occurrence of the Gamma function in the answer to a naturally occurring question, the volume of an $n$-dimensional ball is
$$
V_n(r)=\frac{\pi^{n/2}r^n}{\Gamma\left(\frac{n}{2}+1\right)}.
$$
This example may not be altogether convincing, because the Gamma function can be eliminated in favor of more elementary functions, but it does have the advantage of providing a uniform formula for odd and even $n$, which the more elementary formulas to not. This is just the tip of the iceberg as far as appearances of the Gamma function in mathematics and physics go.
Functions defined in order to compute integrals: If the $\ln$ function were not already known, it would have been necessary to invent it in order to carry out $\int\frac{1}{x}\,dx$. The inverse of the the $\ln$ function would have then led to the exponential function. Similarly, if the inverse trigonometric functions, $\arctan$, $\arcsin$ were not already known, it would have been necessary to invent them in order to carry out integrals such as $\int\frac{1}{\sqrt{1-x^2}}\,dx$, $\int\frac{1}{1+x^2}\,dx$. This would then have led to the creation of $\tan$ and $\sin$.
In the case of elliptic functions, this actually is what occurred. The computation of the arc length of an ellipse (which is of interest in the study of planetary motion) leads to an integral that cannot be expressed in terms of elementary functions. This led to the definition of "elliptic integrals", whose inverses are elliptic functions, to which the theta functions are closely related.
Elliptic functions have the remarkable property of double periodicity. That is, they are periodic in two directions in the complex plane. You can imagine how this might be useful in the study of planar lattices, for example. But more generally, any time you have a function whose domain of definition is the torus, you will find that elliptic functions come into play. Certain complex algebraic curves of degree three are topologically tori; elliptic functions can be used to parametrize positions on such curves, much as $\sin$ and $\cos$ parametrize positions on the circle.
Solutions to differential equations: One of the first classes of differential equations you come to whose solutions cannot be expressed in terms of elementary functions are second order linear differential equations whose coefficients are polynomials of low degree. Hence the Airy functions, Bessel functions, hypergeometric functions, and so on. The simpler the differential equation, the more often it will come up. The differential equations that lead to the above functions arise repeatedly in classical mechanics, electromagnetism, and quantum mechanics.
Orthogonal polynomials: Principles of linear algebra can be brought to bear on the study of spaces of functions. Just as it is useful to write down an orthonormal basis for $\mathbf{R}^n$, it is useful to write down a basis for certain spaces of functions. Depending on the domain and symmetries of the functions of interest, different orthogonal polynomials will come into play. Just as periodic functions may be expanded as Fourier series in $\sin$ and $\cos$, series with other symmetries may be expanded in terms of other types of functions. Often the coefficients of the lowest order terms in the expansion have clear physical meaning. You mention the Chebyshev polynomials. Another example are Zernike polynomials, whose domain of definition is the disk, of which an example is the pupil of the eye. In optometry, the coefficients of some of the low order polynomials relate to certain refractive errors in the eye. In physics and chemistry, the spherical harmonics are used to describe atomic orbitals. They are useful in many other problems with spherical symmetry, including celestial mechanics.
Best Answer
$\mathbb {R,Z}$ etc. are imitating the way we write bold R, Z on a blackboard (hence the name, blackboard bold). It can be argued that when TeXing (not actually writing on a blackboard), you should write $\mathbf {R,Z}$ instead (since that's what $\mathbb {R,Z}$ are meant to represent on a blackboard, in the first place!), and I, for one, do just that most of the time.
They are written in bold to make the name distinct, because $R,Z$ may be used to represent other, more locally defined objects, while bold letters are rarely used as local variables. As to why are the particular letters are used, the $\bf R$ is probably self-explanatory, while $\bf Z$ originates from German (Zahlen).
$\bf K$ as a dummy field name also comes from German (Körper), and in this case bold is likely used to imitate $\bf R,C$ and to indicate that it is "the" background field when it is fixed in the context, so it is, at least locally, as fundamental as $\bf R,C$ are (e.g. in linear algebra and algebraic geometry). It is less often used in that way when we consider many distinct fields ard rings, like in abstract algebra (where letters starting with $K$, and continuing with $L$, and sometimes $M,N$, are still often used to denote fields, but are rarely bolded).