I'm lost now I've got Flash and gone back to my Marauder

[Ecosmith]

Does that mean every class can do the 'invited to the ball!' quest, for example? Is there only one questline for everyone, then? Would a level 15-18 gladiator get invited to a ball in Uldar?

[Gnova]

Does that mean every class can do the 'invited to the ball!' quest, for example? Is there only one questline for everyone, then? Would a level 15-18 gladiator get invited to a ball in Uldar?

No. It is soon after that quest that the main storylines converge and are no longer unique to an area.
The level 1-15 main story questline is unique in each city for characters starting there.
Class quests are available every five leveles for each quest and from 30-50 they may change location when they change from class quests to job quests. I'm sure you know but you have to be holding that class's weapon in order to pickup class quests.

[Niwashi]

Does that mean every class can do the 'invited to the ball!' quest, for example? Is there only one questline for everyone, then? Would a level 15-18 gladiator get invited to a ball in Uldar?

From level 1-15 (up through the quest that makes you an envoy from your starting city to the others) each starting city has its own main quest line, and you can only do the one that corresponds to the city where your starting class's guild is located. Since you started as a Marauder, you'd have the Limsa Lominsan main quest line for those first 15 levels. But then from level 15 on, there is only one main storyline that everyone follows, and it moves back and forth between different regions.