How can I get the girl looking for her fossil to let me pass her so I can get to the second gym?
When you finish super training there is a bar next to your Pokemon's stats. What is the bar for?
You need to find the fossil assistant in Glittering Cave first. Go back through the aquarium to Ambrette Town, and go into the fossil lab if you haven't already (there aren't that many buildings there so try them all until you find it). One of the people in there will tell you to find his assistant. The route that goes to Glittering Cave is right next to the lab.
I don't know - I only answered because I knew the answer to the first question.