Hello Olivier,
Analyzing hw by picture is not ideal, but I can suggest some improvements based on your files.
For USB diff pair you have to keep in mind that the shortest path is the best for signal integrity (and immunity). The picture shows a very long trace for USB data tracks. Keep them short.
It seems to have variable distance between differential tracks. Use the minimum gap your PCB manufacturer can provide and try to not change it during path. Every time you deviate (open or close the gap) you change diff impedance. It is specially important for high speed devices/modes. Don't worry to much in keep 90Ohms diff/45ohms Com impedance. Stress to keep traces parallel and with same impedance all the way. Two vias per trace, as you have, are ok (the ideal is no vias).
Provide a efficient return path. This is the most important for you board. You have a very poor grounding, specially for USB. If you can, in your prototype, increase the grounding using a copper tape. For two-layer board, with low density, it is recommended to ground analog/critical signals and them poor ground on whole board. It will give a good chance of having good return paths for all the signals.
Why does your USB is working after you probe it with multimeter ? I guess you are adding a parasite capacitance, and it is improving your eye diagram. I thought a good layout (diff traces + grounding) will increase the robustness of board.
Regards,
Renato Kiss
