Find the equation of the circle C3 which passes through the center of C1 and through the points of intersection of C1 and C2.

The general equation of a circle is $x^2 + y^2 + 2gx + 2fy + c = 0$, with center $(-g, -f)$.
For $C_1: x^2 + y^2 - 6x - 4y + 9 = 0$:
$2g = -6 \implies g = -3$
$2f = -4 \implies f = -2$
The center of $C_1$ is $(3, 2)$.

The equation of the common chord is $C_1 - C_2 = 0$.

This is the equation of the line $L$ passing through the intersection points.

A circle passing through the intersection of $C_1=0$ and the line $L=0$ can be written as $C_1 + \lambda L = 0$.

Substitute the coordinates of the center of $C_1$, which is $(3, 2)$, into the equation for $C_3$:

The equation of circle $C_3$ is $\boxed{x^2 + y^2 - 4x - 5y + 9 = 0}$.

For $C_1: x^2 + y^2 - 6x - 4y + 9 = 0$:
Center $(h, k) = (3, 2)$.
Radius $r = \sqrt{g^2 + f^2 - c} = \sqrt{(-3)^2 + (-2)^2 - 9} = \sqrt{9 + 4 - 9} = \sqrt{4} = 2$.

The length of the tangent from an external point $(x_1, y_1)$ to a circle $x^2 + y^2 + 2gx + 2fy + c = 0$ is $\sqrt{x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c}$.
For the origin $(0,0)$ and $C_1$:

The length of each tangent is $\boxed{\text{3 units}}$.

Let the equation of a tangent line passing through the origin $(0,