Claiming an unemployed roommate and/or his children for EITC

Your research was correct; there are two types of dependents, "Qualifying Children"(QC) and standard ("Qualifying Relative" in IRS parlance even though they don't have to actually be related). There is no income limit for a QC but there is an age limit, a relationship test and a residence test. Only a QC qualifies the taxpayer for the Earned Income Credit, the Child Tax Credit, Day care credit or Head of Household filing status.

He and his child may qualify as your dependent, but because he is not related, he cannot be a qualifying child for the earned income credit, child tax credit or Head of Household filing status.

 A person can still be a Qualifying relative dependent, if not a Qualifying Child, if he meets the 6 tests for claiming a dependent:

1. Closely Related OR live with the taxpayer ALL year

2. His/her gross taxable income for the year must be less than $4,000 (2015)

3. The taxpayer must have provided more than 1/2 his support

In either case:

4. He must be a US citizen or resident of the US, Canada or Mexico

5. He must not file a joint return with his spouse or be claiming a dependent of his own

6. He must not be the qualifying child of another taxpayer

$4,050 for 2016.

First, persons who are unrelated to you cannot qualify you for the EITC, nor for Head of Household filing status.

That being said, all three of them may still qualify as "Qualifying Relative" tax dependents of yours.  Did he have any income of any type at all, such as unemployment comp? 

Nothing at all.

Then claim all three of them as dependents.  When it asks you for the type, select Other and then Another.