No. This is a common but untrue old wive's tale.
I wish..... No - they don't.
No. No, that's really just a myth. Hair will grow in the way it wants to. Now, it is true that baby hair is very thin and wispy, so after the first haircut you see more of the thicker hairs, but it's not the haircut that made the hair thicker-- it's just that the child is growing.
No. Hair is hair and tends to grow the way it wants to or was meant to regardless of cut. Shorter hair tends to feel thicker and stronger.
No. New hair after a haircut may seem thicker and stronger. That is because the "cut end" of the hair feels coarser. The previous hair ends that were older, and softer/thinner by being "worn down" by friction of day-to-day life, have been cut away. The part of the hair that is outside the hair follicle is not alive, so cutting hair does not change what is happening in the hair follicle in the skin.