Even by missing the level limited quests, you are currently able to get the quest specialist medal on any class regardless of level.
Finally did it! I think I might be the first to do it before 4th job... it was a pain in the ass as I missed out on all the PQ quests and only just had enough with the drumming bunnies, and even buying a htp for the extra 2 quests it gave!
Congrats on achieving Quest Specialist! Unfortunately I cant put your name next to Taniks since the ranking is in order of who achieved it first.
certain ones do, but only the second time you do them. a good rule of thumb is that if it shows up in your available quest tab as opposed to being "available" in the completed quest tab, then it probably will count towards the medal.