In the following score, some of the notes have numbers both above and below the notes. The ones above I assume are fingering advice but what are the numbers below?
Also, what does the III at the start mean?
Alternative fingering advice. The III means to play on the D string rather than the A string, so you'd start off in fourth position with the middle finger, play the first bar in fourth on the D string, switch to third at the start of the second bar and on to second position in its middle. Third bar in fourth position, fourth bar in third. In the sixth bar, you visit the A string for the first time, again in third position (I am assuming that you are not supposed to slide into seventh position: if you really wanted to stay on the D string, fifth position would be sufficient).
This jumping/sliding around on a single string is trying to stick with a single sound quality and to show virtuosity. A typical exhibit for that is the start of Monty's "Csárdás" which typically is played just on the G string for quite a bit.