The moves that Mewtwo can learn in Pokémon HeartGold are Disable, Barrier, Confusion, Future Sight, Barrier, Psych Up, Swift, Miracle Eye, Amnesia, Psycho Cut, Mist, Psychic, Recover, Safeguard, Aura Sphere, Guard Swap, Power Swap, Me First, Ice Beam, Water Pulse, Toxic, Hail, Sunny Day, Taunt, Blizzard, Light Screen, Protect, Frustration, Hyper Beam, Rain Dance, Iron Tail, Solarbeam, Thunderbolt, Thunder, Earthquake, Return, Brick Break, Shadow Ball, Double Team, Shock Wave and Flamethrower.
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double edge, eruption,flamethrower,fire spin
Not without trading from other games. In heartgold you can get ho-oh, lugia, mewto, kyogre, and possibly celebi and jirachi through event. Others can be obtained through trading or other means. It is possible to have all legendaries on one heartgold cartridge, though, if you trade from other game. You need platinum, and emerald, and firered or leafgreen.
All water pokemon!
I don't know all of the Pokemon that can learn Headbutt, but I know that Cubone can learn it.
Gen I: at level 42 to learn all the moves(at 35 it learns flamethrower). Gen II: at 37 to learn all the moves. Gen III: at 41 to learn all the moves. Gen IV: at 47 to learn all the moves. Gen V: at 54 so it learns fire blast(really good).