If one starts puberty at a shorter height, does that necessarily mean that his adult height will be short too? For example, if one started puberty at 5'5 and another started at 4'10, will there be a major height gap, or could the guy who started shorter grow a lot in his spurt and end up similar to the 5'5 guy?
The tendency is for a boy who starts shorter to end up shorter, but it isn't an absolute. As you noted the shorter boy could have a greater growth rate. Let's use your example with some simple numbers. Suppose the shorter boy grows at a rate of 5 inches per year and the taller boy grows at a rate of 3 inches per year. Assuming the growth lasts two years. The shorter boy will end up at 5'8" and the taller boy will end up at 5'11". While the difference in height reduced considerably, in this particular example it did not make up for the large starting difference.