1. $70K is too much income for EIC. $46,000 is the limit for a single parent with one child.
2. Only the custodial parent can claim the dependent care tax credit, even if the non custodial parent is claiming the child as a dependent.* It is a direct reduction of tax due, not an income deduction or exemption. There is no income limit. It cannot be the non custodial parent. The credit is 20-35% of the amount you paid for child care, up to $3000. For someone making $70K, the credit is 20%. So paying $6000 for one child will get you a $600 tax reduction (20% x $3000 = $600).
3. The child tax credit is a straight $2000 per child, under 17, for anyone making less than $200K and more than roughly $15K (it's a complex phase-in). It goes to the parent claiming the child as a dependent**, and not automatically to the custodial parent.
*
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/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)"
**The custodial parent has first priority on claiming the children on her taxes; regardless of the amount of support provided by the non-custodial parent. The IRS goes by physical custody, not legal custody. The non-custodial parent can only claim the child as a dependent if the custodial parent gives permission (on form 8332) or if it's spelled out in a pre 2009 divorce decree. (without conditions - usually the payment of child support). Even if a divorce decree, dated after 2008, gives the non-custodial parent the right to claim the child, he must still get form 8332 from the custodial parent. A properly worded decree should require her to provide that form.
