(i) To find \(\overrightarrow{AB}\), calculate \(\overrightarrow{OB} - \overrightarrow{OA} = \begin{pmatrix} -10 \\ 3 \\ -13 \end{pmatrix} - \begin{pmatrix} 8 \\ -6 \\ 5 \end{pmatrix} = \begin{pmatrix} -18 \\ 9 \\ -18 \end{pmatrix}\).

The magnitude \(|\overrightarrow{AB}| = \sqrt{(-18)^2 + 9^2 + (-18)^2} = 27\).

To find \(\overrightarrow{BC}\), calculate \(\overrightarrow{OC} - \overrightarrow{OB} = \begin{pmatrix} 2 \\ -3 \\ -1 \end{pmatrix} - \begin{pmatrix} -10 \\ 3 \\ -13 \end{pmatrix} = \begin{pmatrix} 12 \\ -6 \\ 12 \end{pmatrix}\).

The magnitude \(|\overrightarrow{BC}| = \sqrt{12^2 + (-6)^2 + 12^2} = 18\).

Since \(|\overrightarrow{AB}|, |\overrightarrow{BC}|, |\overrightarrow{CD}|\) form a geometric progression, \(|\overrightarrow{CD}| = \frac{18}{27} \times 18 = 12\).

(ii) Since \(D\) lies on the line through \(B\) and \(C\), \(\overrightarrow{OD} = \overrightarrow{OB} + t \cdot \overrightarrow{BC}\).

Using the geometric progression, \(\overrightarrow{CD} = \pm \frac{18}{27} \times \overrightarrow{BC}\).

Calculate \(\overrightarrow{OD} = \begin{pmatrix} 2 \\ -3 \\ -1 \end{pmatrix} \pm \frac{18}{27} \times \begin{pmatrix} 12 \\ -6 \\ 12 \end{pmatrix} = \begin{pmatrix} 8 \\ -4 \\ 8 \end{pmatrix}\) or \(\begin{pmatrix} -7 \\ 1 \\ -9 \end{pmatrix}\).