Question 1: It depends what you mean by "easy".
There's a fairly easy construction that works if $K$ has characteristic zero:
Let $K(t)$ be the field of rational functions over $K$, let $\sigma$ be the $K$-automorphism of $K(t)$ that sends $t$ to $2t$, and let $D=K(t)((x;\sigma))$ be the twisted Laurent series ring.
I.e., elements of $D$ are Laurent power series
where $a_i(t)\in K(t)$, with multiplication twisted by the rule $xt=2tx$.
Then $D$ is a division ring with centre $K$. (Actually, replacing $2$ by any element of $K$ with infinite multiplicative order, this also works for fields of prime characteristic that are not algebraic over the prime subfield.)
Chapter 5 of Lam's "First Course in Noncommutative Ring Theory" describes other constructions (or look at Cohn's book on "Skew Fields" for a much more extensive treatment). I think the "Mal'cev-Neumann construction" gives a division ring with centre $K$ for an arbitrary field $K$.
Question 2: I think it's extremely unlikely that a classification is feasible.