(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!