are the end-game dungeons part of the story?

just wondering if the end-game contents such as hm primals and coil are part of the storyline, because i personally am not a fan of end game stuff, but i would really love to keep up with the story, and if possible, i would rather stay away from all the end game rush.

also, will patch 2.1 see an increase in level cap? because if the story is to continue without raising the cap then it's just gonna be a series of lvl50 dungeons and quests all the way like ffxi in 75 cap days?