Hot pants, or 'short shorts', are a fashion item from the 1960s which is still popular today. They first appeared as a design item in so-called 'Swinging London'. The trend for shorter, brighter shorts for young women was started by designers in London such as Mary Quant.
Running or gym shorts are worn by girls as normal wear in some places. Many gym shorts have an inlay made of a comfortable fabric such as cotton, similar to swim shorts (swimming trunks). These are designed to be worn without underwear.
If a person cuts off the legs of a pair of jeans, that's known as a 'cut-off'.