The pay gap is a political tool, it doesn't have any scientific basis. There are way too many variables to account for.
There are studies finding women below 30s earn more then men in the same age bracket. I still don't think there is a pay gap in any direction, every case is individual.
There are several trends the can explain why there are differences in the aggregates (old generations had a more traditional family/work split, old rich men are over represented in the market, more families could survive on one salary in the past, young women have better qualifications than young men, women don't start a family before 30 these days) and there are also tons of biological reasons that make men and women different.
On the subject of paternity leave; while I was grateful to have extra paid time off (it does push your pay per day up), I didn't really enjoy my paternity leave and ended up using it as a work 3 days per week for a long time.
A mandatory paternity solution sounds like a terrible idea for the men who don't want it and it just reduces men's freedom for the increase of a meaningless metric.