What is the probability of generating a given procyclic subgroup in $\mathrm{Gal}(\bar{K}/K)$?

$\hat{\mathbb Z}$ occurs with probability one.

It is easy to define a homomorphism from the Galois group of a number field to $\hat{\mathbb Z}$. To do this, just define a homomorphism to $\mathbb Z_p$ for each $p$. It is sufficient to define such homomorphisms for $\mathbb Q$, as the restriction to the Galois group of a number field will be an open subgroup, also isomorphic to $\mathbb Z_p$. Over $\mathbb Q$, this can be done using the $p$-power cyclotomic tower and the $p$-adic logarithm.

Then by the argument you noted, with probability $1$ the image of your element inside this generates a $\hat{\mathbb Z}$. It is easy to see from this that your element generates $\hat{\mathbb Z}$ in the original Galois group.

For $K=\mathbb{Q}$, this was proved byJ. Ax (Solving diophantine problems modulo every prime, Ann. Math. 85 (1967), 161-183). M. Jarden extended this to arbitrary Hilbertian fields (Algebraic extensions of finite corank of Hibertian fields, Israel J. Math 18 (1974), 279-307). Moreover, he showed that a randomly chosen list of $e$ elements in the absolute Galois group generates a free profinite group on $e$ elements. Global fields, as well as $\mathbb{Q}^{\rm ab}$, are Hilbertian. An excellent source for all this is the book "Field Arithmetic" by M. Fried and M. Jarden.

Regarding the last question, there are non-Hilbertian fields $K$ such that almost all $\omega$ in the absolute Galois group generate $\hat{\mathbb{Z}}$ - see Example 26.1.11 in the Fried-Jarden book (second edition).