Identify the adjacent and opposite sides for angle C.

In $\triangle ABC$, with $\angle ABC = 90^\circ$:
The side adjacent to angle $C$ is $BC$.
The side opposite to angle $C$ is $AB$.

The ratio of $\cot C$ is $\boxed{\frac{BC}{AB}}$.

3.1.2

$AC$ is the hypotenuse.
$AB$ is the side opposite to angle $C$.
$AB$ is the side adjacent to angle $A$.

For angle $C$, the ratio $\frac{AC}{AB}$ is $\frac{\text{hypotenuse}}{\text{opposite}}$, which is $\csc C$.

The trigonometric ratio is $\boxed{\csc C}$.

3.2.1

Given point $P(-3, -4)$, the coordinates are $x = -3$ and $y = -4$. The length $OP$ (radius $r$) is calculated as:

The length of $OP$ is $\boxed{5}$.

3.2.2

From $P(-3, -4)$ and $OP = 5$, we have $x = -3$, $y = -4$, and $r = 5$.

The common denominator for 3 and 25 is 75.

The value of $\sec \theta + \sin^2 \theta$ is $\boxed{-\frac{77}{75}}$.

3.3

To rationalize the denominator, multiply the numerator and denominator by $\sqrt{3}$:

The value of the expression is $\boxed{\frac{\sqrt{6}}{3}}$.