Emerald keeps every evolution method from Ruby and Sapphire. The difference is access: expanded encounter tables and move tutors mean you can complete more chains earlier in the game. Rayquaza is catchable before the Elite Four, and the wider availability of Pokemon gives you more evolution paths to work with.
Same methods, better access. Emerald expanded encounter tables so more Pokemon appear earlier. The Battle Frontier postgame gives additional ways to earn evolution items. Safari Zone also has different spawns. You won't notice any mechanical changes to how evolution works, but you'll have more chains to work with earlier in the game.
Same Beauty-based method as Ruby and Sapphire. Feebas is still found on 6 random tiles on Route 119. The tile locations reset when the trendy phrase in Dewford Town changes. Feed it dry-berry Pokeblocks to max Beauty, then level up. The mechanics are identical, but Emerald's berry blending might feel slightly different due to the different NPC partners.
Yes. The expanded encounter tables mean Pokemon like Vulpix, Duskull, and Roselia appear in more locations. The Magma/Aqua hideout changes also affect which Pokemon you encounter during the story. Overall, Emerald gives you a wider roster to build and evolve through the main campaign.
The Battle Frontier sells rare items for Battle Points, including evolution items like Razor Fang, Razor Claw (in later gens), and some held items useful for trade evolutions. It's a postgame facility, so it supplements your stone and item collection after the main story.
The Elite Four rematches in Emerald give the best experience. Pre-E4, Route 118 and the areas around Lilycove have mid-level wild Pokemon. The Secret Base bliss trick (fighting your own bases with Blissey) is also a classic Emerald grinding method for pushing a Pokemon to its evolution level fast.