Can the noncustodial parent claim the child if the custodial parent takes no part in the US taxes?

The custodial parent's residence, citizenship, and tax status are irrelevant. The ONLY way that you can claim the child is with the 8332 executed by the other parent. That gives you the right to claim the exemption for the child and the Child Tax Credit ONLY. You cannot file as Head of Household, claim the EIC, or claim the Child Care Credit.

Without the 8332 attached to your tax return, your claim will be denied. If you try and claim the child as living in your home that would be treated as tax fraud by the IRS, especially if you claimed the EIC.

if the child lives with the custodial parent in another country, you are in no way able to claim the child, even if you are paying child support,

the child does not live with you, you are not providing a household in the US, Canada or Mexico for the child thus you have no claim and the waiver would be of no effect because she has no standing with the US tax system

As noncustodial parent, you cannot file HOH

YOU NEED signed 8332 ... With SS or ITIN

It permits you ONLY to take child's exemption and CTC

You do not get EITC

Child must have SS#

If the child actually qualifies as your dependent, or you get signed form 8332, sure.

She can get an ITIN.

It has to do with how much you pay for child support. Check the IRS website.

If the mother has an ITIN use that otherwise write "None" and state that she is not residing in the USA.

WHY DO YOU KEEP ANSWERING PEOPLES QUESTIONS BY TELLING THEM THEY ARE A RED FLAG TO GUYS?? YOU'RE A WEIRDO

if the child is not with you, then no

Learn to read, Idiot!

no