When hair is pulled/tugged, does the link between the hair and the follicle become weak? and if so, does the link become strong again?

Weakens over time: If there is constant traction on the hair roots then the follicles do degenerate at a higher rate. Taking the tension away allows roots to strengthen again but over a period of time.

Traction alopecia.: A good example of what you're describing is something called "traction alopecia", wherein people who keep their hair in a tight ponytail start to lose hairs above the forehead. The tight ponytail causes inflammation in the hairline above the forehead, so we tend to pull those hairs out and the inflammation prevents further hair growth. As inflammation subsides, hair can eventually grow back.