Questions in a Character Degradation Model Paper
All g is doing is saying whether the area is closer to a circle (less flat) or a line (completely flat ellipse). 
Yes.
No.  $N(\mu, \sigma^2)$ is a distribution.  You can say 
$$P_k \tilde{} N(\mu, \sigma^2)$$
read as $P_k$ is distributed normal, with mean $\mu$ and variance $\sigma^2$.


To find a value for $P_k$ you need to generate a random number with the mean given by equation (1) in the last screenshot:

$$
\mu = \bar{c}_i + (\bar{b}_j - \bar{c}_i) \left( \frac{d_{ik}}{d_{ij}} \right)
$$

In matlab, this would be something like normrnd(mu, sigma), though what sigma value to choose is not clear from your extractions from the paper.