I want to determine when is this model physically realistic.
I found the stability of the critical points through setting the derivative to 0.
The critical points are v = +/- sqrt(g*m/k).
I have determined the negative point to be unstable since if I plugged in a very negative number into the given equation the derivative would be negative but if I plugged in 0 ( or another number in between - sqrt(gm/k) and + sqrt(gm/k)) it would be positive. I applied the same theory to determine sqrt(g*m/k) is stable.
I also know that when v = sqrt(m*g/k) , terminal velocity is reached. Therefore this model is accurate before terminal velocity. Is my logic here physically sound or is there something I need to consider additionally.
This model is an essential part of classical physics. There's no need to determine whether it is physically realistic. It is realistic in the simple setting of the problem; however, it is not realistic in reality. For instance, the problem assumes the object is a point so you don't have to account for non-uniform shape that interacts with the air differently.
What you did is more in line with stability analysis.