Can a father just take a baby from the mother if they were never married without establishing custody first in Missouri?

No. The unmarried mother has sole custody until the father has established his paternity legally, in court and then requested (and obtained) joint custody and visitations.
No. The unmarried mother has sole custody until the father has established his paternity legally, in court and then requested (and obtained) joint custody and visitations.
No. The unmarried mother has sole custody until the father has established his paternity legally, in court and then requested (and obtained) joint custody and visitations.

No. The unmarried mother has sole custody until the father has established his paternity legally, in court and then requested (and obtained) joint custody and visitations.

No. The unmarried mother has sole custody until the father has established his paternity legally, in court and then requested (and obtained) joint custody and visitations.

No. The unmarried mother has sole custody until the father has established his paternity legally, in court and then requested (and obtained) joint custody and visitations.
No. The unmarried mother has sole custody until the father has established his paternity legally, in court and then requested (and obtained) joint custody and visitations.
No. The unmarried mother has sole custody until the father has established his paternity legally, in court and then requested (and obtained) joint custody and visitations.