Theorem. Every weakly compact cardinal above $\kappa$ is in $S_\kappa$. In other words, if a cardinal $\kappa$ is $\delta$-strong for every $\delta\lt\lambda$, where $\lambda$ is weakly compact, then $\kappa$ is $\lambda$-strong as well.
Proof. Suppose that $\lambda$ is weakly compact and $\kappa$ is $\delta$-strong for every $\delta\lt\lambda$, with $\kappa\lt\lambda$. Let $M$ be a transitive structure of size $\lambda$ with $V_\lambda\subset M\prec H_{\lambda^+}$ and $M^{\lt\lambda}\subset M$. Since $\lambda$ is weakly compact, there is $j:M\to N$ into a transitive set $N$ with critical point $\lambda$. The point now is that while $M$ has all the extenders sufficient to witness that $\kappa$ is $\delta$-strong for all $\delta\lt\lambda$, it follows by elementarity that $N$ has an extender witnessing that $\kappa$ is $\lambda$-strong, and since $V_\lambda\subset N$, it will be right about this. So $\kappa$ is $\lambda$-strong, as desired.QED
I was once at a talk of Sy Friedman, at the logic conference in Wroclaw several years ago, where in a theorem he and Natasha Dobrinen had narrowed the consistency strength of a hypothesis to be between "$\kappa$ is $\lt\lambda$-strong for a weakly compact cardinal" and "$\kappa$ is $\lambda$-strong for a weakly compact $\lambda$," a difference that he had remarked were "within $\epsilon$" and very close together, although they had really wanted to prove an equivalence. I pointed out that actually the two hypotheses were equivalent, by the argument above, and so they had already attained the full equiconsistency they had desired.
The theorem above is can be further improved to weaker related notions than weak compactness, since all we needed was a single embedding $j:M\to N$ with critical point $\lambda$ and $V_\lambda\subset M$ and $M^\omega\subset M$ (in order to know that $M$ is right about the well-founded of the extender ultrapower). This hypothesis is strictly weaker than weak compactness, since every weakly compact cardinal is a limit of such kind of cardinals. It would suffice merely that $V_\lambda\prec_{\Sigma_2} N$ for some transitive set $N$ with $\lambda\in N$.
Update. Regarding your edit, basically anything is possible.
Theorem. If $\kappa$ is $\lambda$-strong and $\kappa\lt\lambda$, then there is a transitive class $M$ in which $\kappa$ is $\lt\lambda$-strong but not $\lambda$-strong, so $\lambda_\kappa=\lambda$ in $M$.
Proof. Let $j:V\to M$ be a $\lambda$-strongness extender embedding for which $j(\kappa)$ is minimal. It follows that $V_\lambda\subset M$ and so $\kappa$ remains $\lt\lambda$-strong in $M$. But $\kappa$ is not $\lambda$-strong in $M$, for if it were, there would be an extender embedding $h:M\to N$ witnessing this, and we could assume that $h(\kappa)\lt j(\kappa)$, since $j(\kappa)$ is inaccessible (and much more) above $\kappa$ and $\lambda$ in $M$; in this case, since $V_\lambda\subset M$ the extender arising from $h$ would have given rise been a $\lambda$-strongness embedding in $V$ with a smaller target for $\kappa$, contrary to the minimality of $j(\kappa)$. QED
So the degree of strongness of $\kappa$ can be singular of cofinality $\omega$, as you desired, or any ordinal at all above $\kappa$.
