You definitely get the level increase with Heavensward, even if you haven't yet finished the MSQ, I did.
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With the expansion you can level up to 60 even if you're behind on the story. Depending on what class/job you are playing there may be some content locked behind ishgard after you cross 50. I was experimenting with the new jobs mostly over the weekend so I didn't explore too much with my current lvl 50 jobs to see what happens beyond 50 for the job progression.
Well all the job quest are available at the get-go. You just start the quest line at 50 but don't get any skills till 52. I'm not sure if you can get them if you don't have HW.
There is level 50 quest for all jobs. To be able to do this you have to finish MSQ 5.5 as well as having the expansion. You can gain level without expansion to 60 but all the new abilities from 50-60 are unlocked - need expansion. I was able to get XP over level 50 without the expansion but cannot get new abilities - since it is quest based.
The lala was quite clear. I only made it clearer for those that misunderstood. This isnt a pissing contest. OP asked a question, and the idea is to give an understandable answer.
Here is the question again, I'd you'd like to reread it. Lemuria asnwered it in a way similar to how it was asked.
You don't need to be in a new area to get exp. My friend got experience from handing in level 45 leves before he even started the new story quests. However you can't get exp from level 50 ARR dungeon mobs...for whatever reason. I guess the devs thought people would just powerlevel by spamming them - or it was too much trouble adding exp to the mobs.
And yes you need the expansion installed to get experience past level 50.