FireRed and LeafGreen remake Kanto's evolution system on the Gen 3 engine. The base game uses Gen 1 methods: levels, stones, and trades. After beating the Elite Four and getting the National Dex, Gen 2 and 3 cross-generation evolutions unlock. Crobat, Espeon, and Kingdra become available through friendship and held item trades.
After beating the Elite Four, fixing the Network Machine on One Island (requires Ruby and Sapphire gems from the Sevii Islands), and getting the National Dex from Professor Oak. Only then can Pokemon like Golbat evolve into Crobat, or Chansey evolve into Blissey. Before that, the game blocks cross-gen evolutions.
No. Even if your Golbat has max friendship, the game won't let it evolve into Crobat until the National Dex is unlocked postgame. Same applies to Espeon, Umbreon, and every other cross-gen evolution. It's one of FRLG's most frustrating restrictions.
Same as Gen 3: walk with the Pokemon, battle with it, use vitamins, don't let it faint. The Soothe Bell (from a woman in Pokemon Tower) speeds things up. Time-of-day doesn't exist in FRLG (no internal clock), so Espeon and Umbreon evolve based on the time your GBA thinks it is, which is normally set during Berry-program patching.
Celadon Department Store sells Fire, Water, Thunder, and Leaf Stones. Moon Stones are found in Mt. Moon and the Pokemon Mansion. The Sevii Islands postgame has additional stones in hidden locations. Pickup ability on Meowth also randomly finds them, same as other Gen 3 games.
Nothing mechanically, but you need a wireless adapter or link cable to trade. The same four Gen 1 trade evolutions work (Kadabra, Machoke, Graveler, Haunter). Gen 2 and 3 held-item trades (like King's Rock for Slowking) also work, but only after the National Dex unlock. You can trade locally or with Ruby/Sapphire/Emerald.