The layers of Kevlar slow the bullet down and catch it. The impact still hurts, but as long as the bullet doesn't penetrate, there is less likelyhood that the wearer will be seriously injured. Vests are very warm and uncomfortable on hot days. They do cut down the wind chill in the winter. Most vests weigh from 3-5 pounds, depending on the amount of bullet resistance and the size of the wearer.
Officers shoot their guns all the time.
We have to train with them regularly to make sure we know how to handle them and are able to hit what we shoot at. By practicing regularly, we will be able to react without having to think about what to do. In a stressful, life-threatening situation, we don't have time to think or make mistakes. The vast majority of police officers have never fired their weapon outside of training. Police officers are given a long list of responsibilities and duties.