You perform the actions stated on a card in sequence. For some spells and abilities, you'll surveil last. For others, you'll surveil and then perform other actions.

When you surveil, you may put all the cards you look at back on top of your library, you may put all of those cards into your graveyard, or you may put some of those cards on top and the rest of them into your graveyard.

Creatures that are going to die after getting -1/-1 will still be on the battlefield while you surveil. They won't be put into their owners' graveyards until after Mephitic Vapors is entirely finished resolving. If any of their abilities trigger on surveilling, those abilities will trigger but won't resolve until after the creatures have left the battlefield.

Mephitic Vapors affects only creatures on the battlefield at the time it resolves. Creatures that enter the battlefield later in the turn won't get -1/-1.