Who can legally claim the child? How?

Your question is not clear. Do you mean claim the child as a dependent for income tax purposes? Your lawyer should have dealt with this question in your Marital Settlement Agreement, and may have done soso check the document. If there is no agreement on this issue, and nothing in the court's orders, then the two parents should work it out between themselves. Generally the parent who provides more than one-half the child's support is entitled to the deduction. If the child lives full-time with the mother, probably it's the mother who is entitled to the deduction. But the two of you can work it out whoever you wish and file a federal form which states the arrangement.

Depends...look at your paperwork/order it should identify who claims the child.

Depends on what the child support order says.

The (physical) primary custodial parent generally claims the child for tax purposes, unless the court's child support order gives the exemption to the non-primary parent, or the other parent signs over that right.