Sorry, guys. After advanced study of probability,
writing a Ph.D. dissertation in applied probability,
and doing applications of probability in US national
security and business, here's my take on the OP:

(1) Sure, given a random variable taking values
in the real numbers, it has a (cumulative)
distribution and may have a probability density
distribution.

(2) In some special, fortunate cases,
actually can get some useful
information about the distribution, e.g., it has
an expectation, the expectation is finite, it has a finite variance, and might
even be able to say that the distribution is in one
of the famous families, Gaussian, exponential,
Poisson, etc. and, then, estimate the parameters
of the distribution, e.g., for a Gaussian, estimate
the mean and variance.

(3) While we know that our random variable has
a distribution, mostly in practice we can't know
much about that distribution. Commonly if we
know that the distribution has an expectation,
that the expectation is finite, and that
the variance is finite (that commonly we get
just by what we know about the real problem
with no effort in probability or statistics),
then we have to be satisfied with those
'qualitative' facts and
move on. E.g., finite variance is enough
about the distribution
to apply the strong law of large numbers!

(4) Mostly when we do know that the distribution
is Gaussian, exponential, etc., it is because
we are using one of the classic limit theorems
of probability, e.g., the central limit theorem
or the renewal theorem.

(5) If the random variable takes values in,
for some positive integer n, some n-dimensional
space over, say, the real or complex numbers, then
in practice knowing much detail about the
distribution is next to hopeless. Why? Because
of the 'curse of dimensionality' where the
number of points needed for a fine grid
is commonly so high it makes 'big data'
look like something on a 3 x 5 index card!

Net, mostly we do our work in applied probability
knowing, of course, that a random variable has
a cumulative distribution but otherwise
knowing at most only some qualitative aspects, e.g.,
finite variance, and little else.

So, our more important methods of working with
random variables should need only meager assumptions
about the distributions of those random
variables. E.g., order statistics are always
sufficient. Commonly to get the most desired
estimates from least squares techniques,
need only finite mean and variance;
the common Gaussian assumptions about
the 'model error terms' is needed only
for doing classic confidence interval,
etc. estimates.

Generally there are a lot of 'non-parametric'
(distribution-free, where we make no
assumptions about the distribution)
techniques in hypothesis testing. For estimation,
there are non-parametric techniques
based on, say, 'resampling' for getting
confidence intervals. And there are other
distribution-free techniques.

Once I saw a paper in hypothesis testing
that was both multidimensional and
distribution-free.

Or, in shortest terms: Yes, a random
variable always has a cumulative distribution,
but that doesn't mean that always in practice
we should try to find that distribution;
instead,
commonly in practice we have little
hope of knowing much about the distribution!

Or, one of the best books in probability
I know of is

Jacques Neveu,
'Mathematical Foundations of the Calculus of Probability',
Holden-Day.

and there is at most only meager mention
of specific distributions in the whole book!