USUM uses the same evolution methods as Sun and Moon with more Pokemon available earlier. Additional encounters mean you can start evolution chains sooner, and new Ultra Beasts add postgame evolution targets. The Alolan form evolution paths are identical to SM.
Level up a Rockruff with the Own Tempo ability between 5:00 PM and 5:59 PM in-game time. USUM made this easier than the event-only method in Sun and Moon. You can find Own Tempo Rockruff through in-game encounters in USUM, no special event needed.
Poipole evolves when it levels up while knowing the move Dragon Pulse. You get Poipole as a gift during the Ultra Beast storyline. Teach it Dragon Pulse via the Move Reminder in the Pokemon League (it doesn't learn it by level-up naturally), then level it once.
The methods are identical. Same locations, same friendship requirements, same level thresholds. The differences are encounter-based: USUM puts more Pokemon in earlier routes, so you can start evolution chains sooner. Dusk Form Lycanroc is also much easier to access without a special event.
Ultra Beasts don't evolve in the traditional sense. But USUM added new Ultra Beasts like Blacephalon (Sun) and Stakataka (Moon), plus Poipole's evolution into Naganadel. These are the only Ultra Beasts with a standard evolution path.
Most of them. Fire, Water, Thunder, and Leaf Stones are available at the Olivia's stone shop in Konikoni City pretty early. Moon, Sun, Shiny, Dawn, Dusk, and Ice Stones show up in Poke Pelago or Isle Aphun once you unlock enough islands. The hardest to get early is the Ice Stone, which requires luck with Poke Pelago mining.