Expected Kolmogorov complexity under Kolmogorov complexity distribution
If $\alpha$ is the answer to the 1st question then $\alpha=\infty$. Namely, for any  $c $  there is an $n $ such that all strings $w $ of length at least $n $ have $K (w) \ge c$. In particular the expectation of $K (w) $ with respect to any distribution on strings of length $n $ is  $\ge c $.

Similarly if $\beta$ is the answer to the 2nd question then $0\le\beta\le 1$, since
$$
(\exists c)(\forall w)(K(w)\le |w|+2\log |w| + c),
$$
but I don't know exactly what $\beta$ is.