Inverter - 7 - Noise Margin Analysis-2 

I guess that is the extreme range of gain. Beyond which the noise gets amplified... (Negative gain because it's an inverter in which a positive change in input value causes decrease in output voltage)

First video of the Inverter - 7, Noise Margin Analysis 1 clarifies that point

because we are here considering short channel and velocity saturation , so have to use Vdsat in current equation.

Because sir is taking slope g at point of (vm,vm) to (0,Vih) slope=g=y2-y1/x2-x1

use trignometry

use y2-y1/x2-x1= slope ie. g as g is slope as well , or g =(vm-vdd)/(Vm-ViL ) similarly proceed for vih thanks :)