Being teetotal, I obviously cannot speak from personal experience on this matter, but I can tell you that your friends are very wrong indeed. Alcohol is a depressant, and caffeine (in coffee) is a stimulant. Being opposites, you can imagine how having your nervous system flushed with both at once can be a problem. Don't consume any stimulants until the alcohol leaves your system. Puking and drinking salt water are both counterproductive in the extreme, because your head-spinning and other hangover effects are due to dehydration. In addition to making you further dehydrated, salt water is toxic and will give you hallucinations should you consume enough of it. Urination will also have no effect, but can be used as a metric to determine how dehydrated you currently are. If your urine is clear, you're fully hydrated. Various deeper shades of yellow signify some dehydration. Orange is extreme dehydration. Brown is liver failure and you're about to die.
The solution to this is to drink water, and a lot of it. Next time you go drinking, have a glass of water in between every alcoholic beverage.