Until end of turn, target creature gains trample and gets +X/+X, where X is the number of creatures you control.
You may cast Authority without choosing any target creatures. Creatures you control will still gain vigilance until end of turn. However, if you choose any targets and each of those targets become illegal before Authority resolves, the spell won’t resolve and your creatures won’t gain vigilance.
While not on the stack, the characteristics of a split card are the combination of its two halves. For example, Destined // Lead is a green and black card, it is both an instant card and a sorcery card, and its converted mana cost is 6. This means that if an effect allows you to cast a card with converted mana cost 2 from your hand, you can’t cast Destined. This is a change from the previous rules for split cards.
Split cards with aftermath have a new frame treatment—the half you can cast from your hand is oriented the same as other cards you’d cast from your hand, while the half you can cast from your graveyard is a traditional split card half. This frame treatment is for your convenience and has no rules significance.
If another effect allows you to cast a split card with aftermath from any zone other than a graveyard, you can’t cast the half with aftermath.
Each split card is a single card. For example, if you discard one, you’ve discarded one card, not two. If an effect counts the number of instant and sorcery cards in your graveyard, Destined // Lead counts once, not twice.
A spell with aftermath cast from a graveyard will always be exiled afterward, whether it resolves, it’s countered, or it leaves the stack in some other way.