@91mommabear -
Q. Does he have the right to claim our child as a dependent?
A. No
Everything you posted is irrelevant except one thing*: the child lives with you more than she lives with the other parent. YOU get to decide who claims the child. If you want to let him claim the child this year, give him form 8332. Without it, he can't claim the child. "He has never given me the IRS form 8332". You are the custodial parent, you don't need that form. He, as the non custodial parent, needs it from you.
This may be helpful in your negotiations with the ex:
There is a special rule in the case of divorced & separated (including never married) parents. When the non-custodial parent is claiming the child as a dependent/exemption/child tax credit; the custodial parent is still allowed to claim the same child for Earned Income Credit, Head of Household filing status, and day care credit. This "splitting of the child" is not available to parents who lived together at any time during the last 6 months of the year; then only one of you can claim the child for any tax reasons. The tax benefits may not be split in any other manner.
Note in particular that the non-custodial parent can never claim the Earned Income Credit, Head of Household filing status or the day care credit, based on that child, even when the custodial parent has released the dependency to him.
So, it's good idea to let the other parent know that you will be claiming those items, as many first time divorced parents are not aware of this rule and may try to claim those items, which will cause the IRS to send out letters.
Ref: https://www.irs.gov/publications/p17#en_US_2017_publink1000170897
Scroll down to "Children of divorced or separated parents (or parents who live apart)"
You can if you are the custodial parent. The custodial parent is (the parent the child lived with for more than 183 days in 2019.
*There may be some question as to the validity in " the agreement he sent to the courthouse"; but the IRS doesn't care about it. They only care about physical custody. The ex cannot present the agreement to the IRS,. The IRS will send YOU all the money. The ex has to try to get the court to give him some of it.
