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.
Yes
A cotton is soft and a cotton jersey is soft Jersey. Cotton is a woven fabric. It is not stretchy. Cotton jersey is a knit. It is stretchy like t-shirt material.
Packing is the placement of gauze or cotton into the nasal area. Packing comes in three forms: gauze, cotton balls, and preformed cotton wedges. Packing is usually coated with antibiotics and, sometimes, petrolatum.
The Cotton gauze I have seen is usually for warmer weather. Those who have worn even "winter" cotton gauze have said it is NOT very warm. All of the cotton gauze that I have seen is only suited for summer wear, unless you are wintering in the Southern Hemisphere or the south when it's having a warm spell. The word gauze automatically makes me think of cotton bandages, which are basically made of thin, see-through fabric and not intended to keep the body warm unless multiple layers are used. I'd check reviews of it on Amazon.com or other clothing sites.
You should use gauze for dressing wounds, rather than cotton balls, because gauze doesn't stick to the wound and leave fibers.
Gauze
Cotton wool is cotton that is cleaned and fabricated into useful items such as gauze for first aid kids, tampons, Q-tips and cotton balls.
Cotton is used to manufacture cotton fabrics and cotton-blend fabrics. Cotton is also used to manufacture other products made from cotton, such as gauze and feminine care products.
Cotton wool is raw cotton with gross impurities removed. It is used for gauze and for tampons. In USA, the term is absorbent cotton: cotton wool is more UK-centric.
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.
Cotton wool is absorbent cotton -- gauze, for example. You may be able to craft something useful or interesting -- from your imagination.
Cotton wool is cotton that is cleaned and fabricated into useful items such as gauze for first aid kids, tampons, Q-tips and cotton balls.