Indeed, 70 can be replaced by any multiple of 10. (10 because the group $I$ (taken as left multiplications) itself already generates 10 points on each circle.)

The reference should be to "Conway and STEIN's book, p. 44, Table 4.1". It comes from the fact that rotations in 4-space can be written as $x\mapsto \bar a\cdot x\cdot b$ for two unit quaternions $a,b$. In 3-space they are $x\mapsto\bar a\cdot x\cdot a$ for a unit quaternion $a$. In this way, every group in $SO_4$ is associated to a subgroup of the (group-theoretic) product of two groups in $SO_3$ (more or less, it's a double-cover). $I\times C_n$ means $I$ is the left group and $C_n$ is the right group. $I$ permutes the 12 great circles in my example. $C_n$ turns each great circle in itself.

I meant Conway & Stein's book on quaternions and octonions.

The term animal or lattice animal is a common term for these creatures (polyominoes, polycubes, etc.) in the physics literature.