First of all, thank you for the adjective "fantastic"(!). The question is actually studied in a paper of mine (link to the MR review). Sorry for talking about my own paper (I have no option, since I do not know if anyone else is interested enough in these questions).
I should add that, in all this, $g\geq 2$. In that paper, what is proved is that any intermediate subgroup either has finite index in $Sp_{2g}(O_K)$ or else contains $Sp_{2g}(\mathbb{Z})$ as a finite index subgroup. In particular, since $Sp_{2g}(\mathbb{Z})$ is a maximal discrete subgroup of $Sp_{2g}(\mathbb{R})$ by the answers to your first question, there are no in between subgroups of infinite index in $Sp_{2g}(O_K)$ other than $Sp_{2g}(\mathbb{Z})$.
