Having just gone through a divorce w/children in CA, I'll share some things I learned.
There are two kinds of custody--legal and physical. They can be awarded as 'joint' or 'sole.' The courts are very hesitant to order sole custody unless there is a valid reason, such as the absence of a parent due to incarceration, or some such thing. Legal custody has to do with who makes decisions about the children, such as where they go to school, etc. Like I said, the courts are quite hesitant to order sole legal or physical custody to one parent unless there is an overwhelming reason to do so.
Then the issue comes down to visitation. Even if you have joint physical custody, that doesn't mean the children have to spend half their time with each parent. My kids spend over 80% of their time at my house, yet we still have joint physical custody. BTW, the percentage of time is one of the factors used in determining the amount of child support. This is probably why he wants it to be 50/50.
If the two of you cannot agree on a parenting plan for custody and visitation and you want the court to make a ruling on these issues, you will be ordered to attend mandatory parenting classes and court mediation. The court mediator's only responsibility is to determine what is in the best interests of the children and make a recommedation to the judge. Judges almost always order exactly what the mediator recommends.
Please get to an attorney ASAP. Especially if you have one in mind. The reason for this is because if an attorney consults with you regarding your divorce, your H cannot hire that attorney. It doesn't matter if you hire them or not, the fact that they talked with you precludes them from talking with him.
PM me if you want.
L