as felonius said, saved minutes do not carry over. There is a reset, but it's not a hard reset. It brings everyone down to 100% fatigue (thus the saved not carrying over), but it doesn't bring everyone up to 100%. It will bring most players up to 100%, but if you have guys that are 97% or below, you probably won't see them at 100%.
And felonius is 100% correct about the first round bye. It's a huge advantage because you do start saving minutes as soon as the 1st round is begun. Nobody knows the exact formula (or if they do, they haven't told), but it's pretty significant.