Crystal uses the same evolution methods as Gold and Silver: friendship, time-of-day, held items, and all the Gen 1 basics. The main difference is encounter availability. Some Pokemon show up earlier or in different locations, which affects when you can start certain evolution chains.
Mechanically identical. Crystal uses the same friendship, stone, trade, and level-based methods as Gold and Silver. The differences are encounter-based: some Pokemon appear in new locations or at different times. The biggest exclusive is the Odd Egg from the Day Care Man, which can hatch shiny baby Pokemon with better odds than wild encounters.
Visit the woman in Goldenrod City's Happiness Checker house (above the bike shop). She gives vague phrases like "It seems to really trust you" or "It's quite cute." There's no exact number display. If she says it's very friendly, you're close to evolution threshold. Otherwise, keep walking and battling.
Yes, but it takes planning. You get Tyrogue from the Karate King in Mt. Mortar after beating him. Its stats at level 20 determine the evolution. For Hitmontop (equal Attack and Defense), you might need to use specific vitamins or avoid certain fights to keep the stats balanced. Save before level 20 and check with the stat screen.
Bill's grandfather on Route 25 gives them out for showing specific Pokemon (same as Gold/Silver). The Schoolboy phone contacts occasionally call to give you stones. Bug-Catching Contest sometimes awards a Sun Stone. The Mystery Gift feature can also produce stones, but that's unreliable.
No exclusive evolution methods or new evolutions. Crystal's exclusives are about encounter changes and the animated sprites. The Celebi event (via the GS Ball) is the biggest unique content, and Celebi doesn't evolve. Every evolution chain works the same as Gold and Silver.