1. No.
2. Yes in effect, but not quite the way you describe.
3. No.
The IRS does not follow state custody orders and they do not evaluate which parent is following or not following orders. The IRS follows where the child physically lives more than half the nights of the year. That is the "custodial parent" for taxes, no matter what the court order says about custody.
Only the custodial parent has the automatic right to claim a child as a dependent. The other parent can only claim the child if the custodial parent signs a release. If the court has ordered you to sign a release and you refuse, the local family court may take action against you, but the IRS will not--without a signed release, the IRS will only award the dependent to the parent who has custody more than half the year.
Even if you sign a release to the dependent claim, the other parent can only get the child tax credit. The ability to claim EIC, the child care credit, and HOH status always stays with the custodial parent and can't be released, waived or shared. In Turbotax, you would indicate the child lives with you more than half the year, but you gave the other parent a signed release. Turbotax will assign the child a status of "Not a dependent, use for EIC and HOH only." If the other parent tries to claim the child as a full dependent (accidentally or on purpose), you may be blocked from e-filing. In that case, print your return and file by mail, and the IRS will eventually get around to investigating the duplicate claim.
Other than the child tax credit and the credit for child care so you can work, there are no special deductions for raising a child. Whether it is horseback lessons or an expensive trip for custody reasons, that's just part of life and not specially deductible.
