Learn from the beginning which battles you want to fight and which ones you will walk away from. If you let your outrage get the best of you, it is real easy to rack up 6 figure legal fees.
Mom is probably unaware that the courts routinely award joint custody if the father wants it.
Request a custody evaluation, the court will likely order it anyway. But if you ask, it adds weight to your desire to be involved. Get a decent atty, not a game player. Suggest Jim Reape, this site. Don't sweat the small stuff, move on to the issues.