Solution:
(1) $2x^2 - 8x = 0$
$2x(x - 4) = 0$
Therefore, $x_1 = 0$, $x_2 = 4$; (4 points)

(2) Given $y^2 - 10y - 1 = 0$,
since $a = 1$, $b = -10$, $c = -1$,
we have $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
$= \frac{10 \pm \sqrt{100 - 4 \times (-1)}}{2}$
$= 5 \pm \sqrt{26}$,
thus, $y_1 = 5 + \sqrt{26}$, $y_2 = 5 - \sqrt{26}$. (4 points)

Therefore, the solutions are $x_1 = \boxed{0}$, $x_2 = \boxed{4}$ for the first equation, and $y_1 = \boxed{5 + \sqrt{26}}$, $y_2 = \boxed{5 - \sqrt{26}}$ for the second equation.