Is there a distribution that describes the number of trials before all the events of a random variable with a discrete uniform distribution occurs?
Yes, there is. I call this distribution the coupon collector's distribution, since it describes the behaviour of the number of trials in the solution to the coupon collector's problem. The general form of the distribution and some of its properties are given here.
Let $X_1, X_2, X_3, ... \sim \text{IID U} \{ 1,...,m \}$ be uniformly distributed over $m \in \mathbb{N}$ categories and let $T$ be the smallest number of values in this series required to cover all the categories. This random variable follows the coupon collector's distribution, which has mass function:
$$\mathbb{P}(T=t) = \frac{m!}{m^t} \cdot S(t-1, m-1) \quad \quad \quad \text{for all } t \geqslant m,$$
where the function $S$ denotes the Stirling numbers of the second kind. It can be shown that this distribution has cumulant generating function:
$$K_T(s) = \ln((m+1)!) + sm + \sum_{r=0}^{m-1} \ln (m-re^s) \quad \quad \quad \text{for all } s \in \mathbb{R}.$$
Its mean and variance are given by:
$$\mathbb{E}(T) = m H_{1,m} \quad \quad \quad \mathbb{V}(T) = m^2 H_{2,m} - m H_{1,m} \quad \quad \quad H_{k,m} \equiv \sum_{i=1}^\infty \frac{1}{i^k},$$
where the values $H_{k,m}$ are the generalised harmonic numbers. For large $m$ the harmonic numbers in the expression for the mean are often approximated by their asymptotic form to give $\mathbb{E}(T) \approx m (\gamma + \ln m)$ where $\gamma$ is the Euler-Mascheroni constant. (The latter expression is sometimes incorrectly given as an exact expression for the mean.)
Also, is there a generalized distribution to describe the number of times needed for all events to occur at least $K$ times.
Such a distribution certainly exists, but its form is quite complicated and cumbersome. For this latter distribution it is usually simplest to deal with it by simulation (see e.g., this answer to a related question).
