An action to determine paternity can be started by the mother, the potential father, a guardian or conservator of the child, or the state. Paternity actions receive priority over other cases. Blood tests can be required by either party or at the court's request. Very accurate blood tests now exist. If the question isn't settled, a trial will be held to determine paternity (ARS §§25-801-25-817).

Yes, but they are not absolute.  In a proceeding for dissolution of marriage, paternity, or modification a court will award determine custody according to the best interests of the child, which may sometimes be joint custody or sole custody with one parent. (A.R.S. § 25-403.01)., and may denyThe court will also order parenting time to the non-custodial parent unless visitation to thewith  non-custodial parentnon custodial parent  if iItwould seriously endangers the child’s physical, mental, moral or emotional health (ARS § 25-408). 

Finally, with any custody order the court will determine an amount of child support. Even if the parents enjoy a joint custody arrangement, there is still a responsibility of either parent to provide for the support of the child. (ARS § 25-403.09).

Parental rights can also be terminated by the court when:

• The parent has abandoned the child.
• The parent has neglected or willfully abused a child.
• The parent is unable to carry out parental responsibilities because of mental illness, mental deficiency or a history of chronic abuse of drugs or alcohol.
• The parent is deprived of civil liberties due to the conviction of certain felonies.
• A potential father has failed to file a paternity claim or action in certain circumstances (ARS §8-533).