We pay more attention to teenagers, but adults make the final decision. Depending on the age of the child, such factors include scheduling, consistency of care, time commitment, keeping appointments, e.g., pediatrician, safety, environment, etc. Generally yes, especially when small children are involved. However, there are many exceptions, especially if the mother is unfit. The father has a much better chance, especially with teenage boys.
Many judges believe that a full time father is necessary to maintain discipline with teenagers.
Instead of Parenting Time, the child lives with both parents, usually equally. If one parent has a higher income, expect a transfer of some funds to the other parent. The support, however, should be lower than an order determined by the child support guidelines. No, not without a court order. File a contempt action to collect child support.