Technically access to the roulette is unrelated to your level. You can be level 50 and not have access to the low level roulette if you haven't unlocked Tam-tara.
I think the requirement for the low-level and guildhest roulettes is that you have access to at least 2 of the dungeons. There has to be some uncertainty which you will get. So if you have unlocked Tam-tara on a different class the low level roulette will then become available when you reach the minimum level for Tam-tara.
For main story, hard, expert, and trial roulette I think you have to have unlocked ALL of the included dungeons. After the patch my trial roulette was locked until I progressed the storyline enough to do Ramuh.