We present eigenvalue estimates of integral operators associated with compositional dot-product kernels. The estimates improve on previous ones established for power series kernels on spheres. Our estimates allow us to obtain the volumes of balls in the corresponding reproducing kernel Hilbert spaces. We then discuss the consequences in statistical estimation with compositional dot-product kernels and highlight interesting trade-offs between the approximation error and the statistical error depending on the length of the compositions and the smoothness of kernels.