Decidability of $\lbrace \langle D \rangle \mid \text{$D$ accepts $a^kb^k$ for some $k > 0$}\rbrace$
Hint: Let $s_0$ be the starting state, $F$ the set of accepting states, and $q$ the transition function. The mapping $k \mapsto q(s_0,a^k)$ is eventually periodic and can be determined explicitly. The predicate $(s,k) \mapsto q(s,b^k) \in F$ is eventually periodic and can be determined explicitly. So the predicate $k \mapsto q(q(s_0,a^k),b^k) \in F$ is eventually periodic and can be determined explicitly.

Similarly, you can check whether a DFA accepts some word of the form $a^k b^k c^k$. The fact that the languages $\{ a^k b^k \mid k \in \mathbb{N} \}$ and $\{ a^k b^k c^k \mid k \in \mathbb{N} \}$ are not regular doesn't play any role here.