Picking up the Sledge is a level 50 quest, so you can't do it on a class below that level.

As for your dragoon level boost, there could be a variety of explanations. You don't get the levels instantly. You are instead sent an item called "Tales of Adventure: One Dragoon's Journey", which you can use to instantly boost your dragoon to level 60 and mark all job quests up to that point as complete. Check to see if you've simply forgotten to use that item.
If you for some reason haven't recieved it at all, I'd recommend deleting any system mail from your mailbox, then pressing the request delivery button, as there is a limit to the number of system mails you can have at a time.