The custodial parent is unemployed and on full disability. She is also married but legally separated and not living with a man who is not the child's father. Can the non-custodial biological father, who pays child support and gets the child every weekend claim the child without a signed form 8332? Can the legally separated husband who is not the child's father, does not pay support, does not live with the child or spend any time with the child, can he claim the child?
No, as the Non-Custodial Parent, you will need a signed Form 8332 from the Custodial Parent of your Child in order to claim them.
In addition, your Ex-Wife's legally separated husband cannot claim the child as a Dependent unless he receives a signed Form 8332 from her as well. If they are still legally married, per IRS rules, the child is his legal child also.
So basically, if your Ex-Wife is the Custodial Parent of your Child, she can determine who can claim the Child if she chooses not to by filling out Form 8332 and giving it to that person to attach to their Tax Return.
You can check our Form 8332 & the Instructions here: https://www.irs.gov/pub/irs-pdf/f8332.pdf.