When the pan is cast iron like the one in the picture, the hot pan transmits heat directly into handle as they are all one piece. How fast the handle gets hot depends how long the pan has been heating and how hot you are cooking. If you’re just doing a quick fry or sear, you’ll be fine, if you’re simmering something for a while, you’ll need protection.
2. When you use a gas stove, a lot of heat is wasted* as hot air that flows up and around the pan. If the handle is metal and close to the pan or not designed with sufficient heat isolation and dissipation, it will get very hot.
*Some cookware, like woks, capture and use some of this heat, with their wide sloping sides, but most of it still escapes unutilized. In periods of cold weather, excess heat that is not vented helps to heat the space. Otherwise the heat needs to be vented, or offset by running air conditioning.