How can I increase my reputation? I would like to do the harder missions but I can only do easy since I apparently don't have a high enough reputation to do medium or higher. Does it slowly increase with each mission I do?

If my memories are still correct it increass each time you kill an ennemy in that area of the map or when you compleat missions successfully. ^-^

It also increases if u participate in her aid events (yellow banner you can see from the world map at certain hours)