If someone begins puberty early, will this mean that they will be shorter in the end?

Possibly. The earlier one starts puberty, the sooner the growth plates close. However, this doesn't necessarily imply that the person will be short. It only means that the individual will stop growing sooner.
May be. It all depends on the height the start of puberty, how advance into puberty the child is and the parental height.
Yes and no. Puberty is more a transition with a variable speed and intensity than signal of the end of growth. Some kids move quickly thru stages in a few years & some take about 7. The pattern of puberty often follows that of a parent but eventual linear growth is defined more by bone maturity than age or pubertal signals. A bone age study & growth records can give you an idea how of how tall a kid will get.