In this paper, we establish a compactness criterion for the composition-differentiation operator \( D_\Phi \) in terms of a decay condition of the mean counting function at the boundary of a half-plane. We provide a sufficient condition of the boundedness of the operator \( D_\Phi \) for the symbol \( \Phi \) with zero characteristic. Additionally, we investigate an estimate for the norm of \( D_\Phi \) in the Hardy-Hilbert space of Dirichlet series, specifically with the symbol \( \Phi(s) = c_1 + c_2 2^{ -s} \). We also derive an estimate for the approximation numbers of the operator \( D_\Phi \). Moreover, we determine an explicit conditions under which the operator \( D_\Phi \) is self-adjoint and normal. Finally, we describe the spectrum of \( D_\Phi \) when the symbol \( \Phi(s) = c_1 + c_2 2^{ -s} \).
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