Materials (i assume you mean clothes of fabrics) are stretchy due to the amount of elastic in the fabric. More elastic means stretchier fabric, less elastic means it will not be as stretchy.
stretchy pants
yes there is a panal in the front of pants that is a thin breathable stretchy fabric
clothes are made of fibers woven together that air can slip between, but something called a zentai suit, its airtight body suit that's very stretchy and you can inflate in that.
strong rope, used for parachutes and stretchy
For baby cothes, you want to use anything soft and stretchy and, if possible, flame resistant. I would suggest cotton knits with or without a bit of polyester to help hold the shape.
elastic (a stretchy string)
Cotton itself is not a stretchy fiber. Woven fabric is typically not stretchy. So no, I would not expect a cotton gauze to be stretchy unless it was a cotton blend.
Dark colors have a slimming effect. Another answer Sweat pants or any stretchy spandex type material.
They are somewhat stretchy.. But don't force them to stretch further then they are supposed to go.
Collecting Used Clothes
No, it is made stretchy by either chicle or sometimes (ew) rubber.