Length of Stay (or Ultimate Park Hopper as they are called now) gets you into DQ as well as the water parks, PI. You have to use your plusses to get into PI and Waterparks. No DQ with park hoppers.
If you buy your UPH with your package, I think you can get a small discount on it, but not drop days. It starts the day you arrive and ends the day you leave. If you get it when you check in, you pay full price, but they might let you drop the first or last day (sometimes both).
Hoppers and their plusses are good forever until you use them up. UPH is only good while you are there for that trip. If you change resorts, you may have to purchase another one for that length of stay.
For the hopper, every day you enter a park is a day used. But if you only go to a water park or PI with a plus and none of the four parks (MK, AK, DS or Epcot), you don't use a day of your hopper pass. You can go from park to park on the same day.
UPH will be on your room key. Regular PH with or without plusses might not be on your room key.