How has the rise of limited formats affected MTG set design?
Can't answer number one but there have been a number of articles published regarding number 2. When designing sets Wizards takes a serious look at the rarity of each card and its effect on limited. Its okay for a mythic rare to be completely, off the wall busted in limited (though they do try to avoid it) because the odds of pulling that card will be so low that it won't have much effect on any given limited tournament. Odds are even in a tournament with 16 people and 3 packs per person that card might not show up at all. The odds are even lower that that specific mythic will show up more than once and even lower still that the same person will be able to get it more than once.

Uncommons and especially commons on the other hand are a different animal. Its a common occurrence in drafts for players to have multiple copies of common and an uncommon cards and for multiple players to have copies of those cards thus the power level of these cards in relation to the rest of the limited environment needs to be more closely tested. Similarly because the bulk of all limited decks will be commons and uncommons wizards needs to make sure of a couple of things.

First they need to make sure that certain roles are filled by the commons and uncommons. This is the reason there will always be some form of common creature removal, common mana fixing (espescially true in a multi color oriented set) and likely some form of common draw and common non-creature removal. It is also one of the reasons that every set will contain common red burn, common blue draw, common green creature pump, common white weenie cards, common black discard ect.

If the set has a new theme or mechanic to it, as most sets do (I can't think of any in recent years that don't) Wizards also needs to make sure those themes and mechanics appear in some form or fashion at the lower rarities and that there is synergy among the cards at those rarities.