The story in Guild Wars 1 is more about the races' individual struggles and the balance of power among them. Guild Wars 2's story is more focused on a common threat, the Elder Dragons. Guild Wars 1 has a more fragmented story in terms of different regions having their own issues. While in Guild Wars 2, everything is more connected due to the overarching dragon threat.
Yes, level boosters in Guild Wars 2 generally skip the story. They are mainly designed to quickly boost your character's level, so you can jump ahead in terms of character progression without having to go through the story content that would normally be associated with leveling up in the traditional way.
After the story, you might want to focus on PvP battles to test your skills against other players. Or, try your hand at crafting and improving your equipment.
Yes, usually you need to pay for Guild Wars 2 Living Story. It often comes with expansions or special purchases. The Living Story content helps to expand the game's universe, introduce new quests, characters, and areas. Buying it enhances the overall gameplay experience.