Turned 19 in summer after graduating Home High School
Take a look at IRS Publication 17 page 27 at https://www.irs.gov/pub/irs-pdf/p17.pdf. In the table under "Tests to be a qualifying child" item #2 states "The child must be (a) under age 19 at the end of the year and younger than you (or your spouse, if filing jointly); (b) under age 24 at the end of the year, a student, and younger than you (or your spouse, if filing jointly); or (c) any age if permanently and totally disabled."
If your child was a student for at least 5 months in 2021, this requirement is met. If they "graduated" on May 1st, then the entire month of May counts.
As for the EIC, since your child turned 19 in 2021, they don't qualify you for the EIC. They have to be "under" the age of 19 at the end of the tax year in order to qualify. See https://www.irs.gov/credits-deductions/individuals/earned-income-tax-credit/qualifying-child-rules#T...
