What do you do when you don't have a discard during the game? Also, what happens when you lay down a skip card? Does the color change to what the next player lays down or do you have to use the color played before the skip card was played?
If you don't have a discard during the game, it means you have played all of your cards, so the round ends. You can end the round either by discarding your final card, or by playing all cards in your hand.
The color doesn't change in Phase 10. Perhaps you are thinking of Uno rules? You can discard any card you like; the color doesn't matter.
In my experience if you don't have a discard, you're out of the game. Some variations allow you to still compete for snaps though.