Father lied on coustdy papers?

My baby's dad had his girlfriend serve me and at that she was rude and gave my daughter a dirty look(she doest like us cuz where part black). I went through the paper work and he lied saying he paid for her birth, which i did cause i can prove it through my insurance and the rest i had to pay for. He then said he pays for the home he lives in but his dad just bought it. and that no one in his family works which again i can prove is a lie. He then wants joint custody and wants visitation every week for two days where she'll spend the night. Can the lies benefit me in court and can it help me prove he's an unfit father and a constant lier?