You have asked two questions that have different answers.
Some materials, like plastics, are flexible in part because of a property called ductility. Essentially, they are stretchy. Plastic molecules are long chains and can flex to absorb stress. However, this property is only effective above a certain temperature, called the “glass transition temperature.” Below that temperature, the material loses that flexibility and becomes brittle. At that point, it can no longer absorb the stress in the same way, and instead shatters.
Your body does a variety of things to cope with the cold. For instance, it redirects blood flow away from the extremities (hands and feet) by narrowing the blood vessels, in order to try to keep your core organs warm. This can make your muscles feel stiffer and less responsive. Skin can get dried out, more rigid, and more sensitive. The body may interpret nerve signals from the skin that are complaining about *both* pain and the cold as pain. For all of these reasons, you may feel more pains or aches while cold.