The screen of a modern LCD is a technically complex beast. A grid is laid across the screen and at each intersection of this grid lies a pixel. The number of pixels on the screen will dictate the resolution of your screen. A 1080p screen uses 1,920 pixels across the horizontal plane and 1,080 pixels across the vertical plane; resulting in over 2 million pixels.
Dead pixels are a common issue with TV sets. Sometimes the conductor or transistor behind the pixel will die out and will no longer be able to light up that pixel. A few dead pixels may be annoying but are not worth worrying about.
Prognosis: Good. Annoying, but not a sign of imminent death. Some warranties will specifically say they will only fix TVs with a certain percentage of dead pixels.
Like a dead pixel, a scratch in the screen will not likely result in any major issues - other than annoyances. Repairing a scratched screen is difficult and has the chance of making the screen worse. Your best bet is to turn on the TV, turn on a TV show or movie and see if you can notice the scratch - in many cases the scratch will only be noticeable if the TV is off. If the scratch doesn't bother you while watching a show, don't worry about it! Many products claim to be able to fix scratches, but these are usually junk products trying to make a quick buck.
Prognosis: Good - again, the scratch may be annoying but in many cases won't be noticeable when a show is on.
Oh boy.... you have a broken screen. The prognosis will depend on the type of break. Does the TV still function? If yes, you're in luck. If not, there isn't going to be much you can do. Screens are notoriously expensive to replace and typically can't be performed by the end consumer. It's definitely worth getting a quote from an authorized repair center but don't hold your breath.
Prognosis: If the screen doesn't work anymore I'd contact a repair center but get your wallet ready. Keep in mind TV prices continuously drop so if you had a $3000 TV from 5 years ago, you can probably upgrade to a better TV today for half the price.