Well, as far as I know, the producer's first skill is for, well, production, but the second one, influence ... I can't help but wonder what it does. Because as far as I saw, it doesn't change anything on drop rates in the auditions (but holding audition make you level up the influence skill), it doesn't make you win more friendship/influence or romance with the idol when you plan date with them ... So I really wonder what does it do ?

I remember reading about producer influence in regards to snatching idols from other groups a while ago. If I remember correctly, among other factors, the higher your producer's influence level, the higher the chance of you successfully convincing an idol from another group to join your own. Keep in mind that this has not been implemented into the game yet, and was/is part of the planned asynchronous multiplayer features. Not much has been said about such features since the end of the Kickstarter campaign, however, so this may be subject to change.

Other than that, all I can see it doing is maybe how easy it is to convince idols to postpone their graduations and similar scenarios, regardless of how much personal influence you may have over than individual person.