Let's break down and solve the problem step by step.

We are given the complex number $z = -i$, where $i = \sqrt{-1}$. The goal is to compute the sum of various arguments:

$\arg(z) + \arg(\overline{z}) + \arg(|z|) + \arg(\arg(z)) + \arg(\Im(z))$

Step 1: Calculate $\arg(z)$

We know that $z = -i$. In polar form, the argument of $-i$ corresponds to the angle made with the positive real axis. Since $-i$ lies on the negative imaginary axis, the argument is:

$\arg(z) = -\frac{\pi}{2}$

Step 2: Calculate $\arg(\overline{z})$

The conjugate of $z$, denoted by $\overline{z}$, is $\overline{-i} = i$. The argument of $i$, which lies on the positive imaginary axis, is:

$\arg(\overline{z}) = \frac{\pi}{2}$

Step 3: Calculate $\arg(|z|)$

The modulus $|z|$ of $z = -i$ is:

$|z| = 1$

Since $|z|$ is a real positive number, the argument of any positive real number is 0:

$\arg(|z|) = 0$

Step 4: Calculate $\arg(\arg(z))$

We previously found $\arg(z) = -\frac{\pi}{2}$. Now we need to compute the argument of this real number. Since $-\frac{\pi}{2}$ is a negative real number, the argument of any negative real number is $\pi$:

$\arg(\arg(z)) = \pi$

Step 5: Calculate $\arg(\Im(z))$

The imaginary part of $z = -i$ is $\Im(z) = -1$, which is a negative real number. Again, the argument of any negative real number is $\pi$:

$\arg(\Im(z)) = \pi$

Step 6: Sum all the arguments

Now, we sum all the computed arguments:

$\arg(z) + \arg(\overline{z}) + \arg(|z|) + \arg(\arg(z)) + \arg(\Im(z)) = -\frac{\pi}{2} + \frac{\pi}{2} + 0 + \pi + \pi = 2\pi$

Step 7: Express in the form $\frac{p}{q} \pi$

We have $2\pi$, which can be written as:

$2\pi = \frac{2}{1} \pi$

Here, $p = 2$ and $q = 1$. Since $p$ and $q$ are coprime, the sum $p + q$ is:

$p + q = 2 + 1 = 3$

Thus, the value of $p + q$ is 3.

Would you like further details or have any additional questions? Here are some related questions to expand your understanding:

1. What is the geometrical interpretation of the argument of a complex number?
2. How do you compute the modulus of a complex number in general?
3. What is the argument of the conjugate of a complex number, and why?
4. How does the argument behave for purely real numbers, both positive and negative?
5. Can the argument of a complex number ever exceed $\pi$ or be less than $-\pi$?

Tip: When dealing with complex numbers, always remember that arguments are taken in the principal range $(-\pi, \pi]$.