Yellow uses the same evolution methods as Red and Blue: levels, stones, and trades. The roster is identical at 151 Pokemon, but your starter Pikachu refuses to evolve into Raichu. You get all three Gen 1 starters for free during the story, which changes which evolution chains you can complete early.
No. Your starter Pikachu refuses the Thunder Stone. It won't evolve into Raichu no matter what. This matches the anime where Ash's Pikachu never evolves. If you want a Raichu, you'll need to trade one in from Red or Blue.
NPCs give them to you during the story. Bulbasaur comes from a woman in Cerulean City (needs high Pikachu happiness). Charmander comes from a trainer on Route 24. Squirtle is given by Officer Jenny in Vermillion City after beating Lt. Surge. All three can be fully evolved in one playthrough.
The methods are identical: levels, stones, and trades. Your starter Pikachu's refusal to evolve is the only exception. Some Pokemon appear in different locations and at different levels, which affects when you can start certain chains, but nothing about the evolution mechanics themselves changed.
Yes. Yellow is fully compatible with Red and Blue for trading and trade evolutions. Link cable works between all three versions. You can trade your Haunter to a friend's Red cartridge and get Gengar back. The only thing that doesn't transfer well is Pikachu's happiness data.