1. Are young stars are made of dust and gas and old stars are made of dust only?
Stars are mostly composed of hydrogen and helium, with just a few percent of everything else: water, carbon, iron, etc. The "everything else" comes largely from interstellar dust, so technically we can say (though we usually don’t) that all stars are composed of gas and dust. However, stars are so hot that any dust included in the star was likely vaporized during its formation. Thus, all the material in any star is now completely gaseous.
2. How are the distances to other stars measured accurately?
The distances to "nearby" stars (within a few hundred light-years) are measured using parallax. A light-year is the distance that light travels in a year, 9.5 trillion kilometers (about 63,240 AU). The Sun is 8.3 light-minutes away and the next nearest star, Proxima Centauri, is 4.2 light-years away. Below is a diagram illustrating parallax.