The right of the father is basically to have parental decision making and parenting time adjudicated in the form of a custody order and child support order the context of a judicial proceeding, and to have the orders entered in the judicial proceeding, typically, in the case of a married couple in the form of a legal separation or divorce proceeding.
Both parents are also obligated to adhere to all court orders entered by a court with jurisdiction over the parties. Most of the "hard law" in custody cases involves jurisdiction, enforcement of orders, and child support calculations. There are also guidelines on how often a custody order can be modified absent an emergency exigent situation.
A court making custody orders is "sitting in equity" which means it has broad discretion to fashion an order and that there is no one right or wrong answer. A wide variety of rulings would be upheld on appeal. Two judges could enter extreme different custody orders with extremely different amounts of time with each child and each one would be upheld on appeal.
Once a custody case is put in a judge's lap, even a complete settlement between married parents short of mutual voluntary dismissal of the entire case, is not binding without also receiving court approval, because the judge must consider the child's interests as well as the mutual agreements of the parents, although those mutual agreements are usually honored.
A judge can even order some parenting time with someone who isn't a parent but has a relationship with the child, so long as each parent can maintain some sort of parent-child relationship with the child.
For example, a judge could order that the child leave most of the time in an aunt in Denver, while mom lives in Seattle, and dad lives in Phoenix, while only visiting parents on alternating extended school holidays, or in the Denver area.
The judge has extremely wide discretion applying a best interests of the child standard to establish parental decision making and parenting time. So long as each parent has enough regular parenting time to maintain an ongoing relationship with the child, probably for at least a number of nights every year, as opposed to de facto having parental rights terminated, anything goes if the judge supports a ruling with evidentiary findings of fact which are given great deference.
A custody order can micromanage most aspects of daily life that actually involve the child, but not matters that do not involve the child (e.g. a judge can't order a parent not to date because that will be harmful to the child and cannot order a parent to continue to live in a particular town).
One commonly applied theory of applying the best interests of the child standard is to make an effort to roughly approximate the pre-dispute status quo insofar as it is feasible to do so for separated parties. But judges have extreme wide discretion. Certainly, neither parent has a right to a specific number of nights per year, to an approximately equal amount of total parenting time, to parental decision making on any particular issue, or to in person visitation every single month.
Restrictive conditions such as supervised parenting only can be imposed, but are generally only imposed for good cause.
There is a special body of case law specifically pertaining to custody arrangements in connection with an interstate move by a parent from a common residence. To oversimplify, the basic bottom line in that case law is that there are no strong presumptions one way or the other in that situation, and that it must be handled on a case by case basis in light of the facts and circumstances.
The age and needs of the child, the interactions of parent and child, the ability of a parent to personally provide child care as opposed to doing so though family members and/or baby sitters, the ability to provide a decent home environment economically, the cost and logistics of exchanging custody between states, etc. all play into the analysis. Often there would be some element of insuring communication between a non-custodial parent and the child on a more regular basis.
The older a child is, the more the child's own views, communicated to the judge or a court visitor or evaluator, matters. With younger children there is a preference for more frequent contact if feasible. With older children, custody arrangements where a child lives mostly with one parent and visits the other during extended school holidays are common.
As a practical matter, with a young child, it is highly desirable for a plan to be mutually agreed as causing compliance via court order and process is expensive and frequently makes things worse rather than better. Judges expect a certain level of flexibility and tend to punish parents who are too rigid, but also tend to punish parents who defy court orders.
Of course, absent a custody dispute, a parent has broad authority over their child in circumstances that don't amount to abuse or neglect of a child and don't violate a court order under a notion called being a "natural guardian" of a child. But this is not defined in either statutes or case law, with any great specificity.