How important is it to use the correct fingers when playing?
I think that comfortable and efficient fingering is very important—but the suggestions offered by an editor aren't necessarily the best ones for everyone. (If you compare fingerings in different editions of the same piece, you'll quickly see that there's often no universal consensus of what's "correct" among editors.) In my opinion, the best approach is to make careful choices but keep in mind that what works in slow practice may not work so well at tempo. When you practice, always follow your fingering decisions consistently (but don't be afraid to modify a fingering that turns out to be clumsy).
Also - what is the best way to advance from one level to the next higher level?
No matter what ranking system you're following, there are no abrupt demarcations between levels. Every piece is different in its challenges, and all categorizations by difficulty can be expected to be relative and somewhat arbitrary. If you want to play increasingly advanced music, choose new pieces that are a modest stretch for you; your reach should exceed your grasp, but not by too much or you'll likely wind up frustrated (at the very least).
Topic: fingers numbered