1. Your niece (the taxpayer's daughter), can be claimed as a dependent if her mother (your sister) paid more than half her support for the year and your niece had less than $4150 of taxable income. If the taxpayer is not sure if she paid half her daughter's expenses, there is a worksheet here. https://www.irs.gov/forms-pubs/about-publication-501 Your niece would file as single and check the box for "I can be claimed as a dependent by another taxpayer."
2. The baby (the taxpayer's grandchild) can be claimed as a dependent by the grandmother if all the following statements are true:
a) the baby lived in the grandmother's home more than half the year, or more than half the days since she was born if born in 2018,
b) neither the child's mother or father will claim the baby as a dependent (both parents agree to let the grandparent claim the child),
c) the grandmother has more taxable income than either the baby's mother or father. (Depending on your sister's age and situation, remember that social security is not taxable.)
3. The boyfriend can't be a dependent of your sister. His income disqualifies him.
4. The boyfriend OR the niece could claim the child as a dependent if they wanted to, as long as the baby lived in the same home with its parents more than half the year, or more than half the days since she was born if born in 2018. Who would get the biggest refund for claiming the baby (mother, father or grandmother) depends on who has the highest taxable income. While the father or mother would qualify for some EIC, they would not get the full child tax credit. It's probably best for the grandmother to claim the baby as a dependent, but to get an exact figure, the different scenarios would have to be tested.
5. Whoever claims the baby as a dependent can also file as head of household instead of single, but only if they are the person who pays more than half the total household expenses. (My guess is this is the grandmother.) Everyone who can't file as head of household files as single. If the baby is claimed as a dependent by someone who does NOT pay more than half the household expenses, then everyone files as single (unless your sister is married and you didn't mention it).
6. The boyfriend might be able to claim your niece (his girlfriend) as a dependent, if he paid more than half her living expenses for 2018. This may require some calculations. The boyfriend and grandmother can't both pay more than half the expenses of the mother. But if, for example, the grandmother paid the baby's expenses but the boyfriend paid his own and his girlfriends expenses, it might work out that way.
I suspect that in the end, the grandmother claims both the grandchild and her adult daughter, and the boyfriend stands alone. But all the fact must be considered. All these rules are covered in publication 501.
7. I have no idea how any of this will impact the child's qualifications for state insurance.
