can i claim her on my taxes this year?

To be a dependent is not dependent on how much time, but the fact that you have that for even 1 day of the year.

Just like a couple who has a baby born on December 31, at 11:59, as long as the birth certificate shows that the child was born in that year they can claim it as a dependent and so you in this case.

"Say Bye" is completely wrong. The "couple who has a baby born on December 31, at 11:59" is allowed the claim only under a very specific rule that states that it applies only to a child who lived with you for the entire portion of the year that it was alive. Since it did not live with year for the first six days that it was alive, that rule technically does not apply. Since you did not have her the entire portion of the year that she was alive, and you did not have her more than half a year, she is technically not your "qualifying child". However, if she is not a qualifying child of anyone else, then she might be your "qualifying relative".

I would claim the child as your dependent. You could have issues with her mother but that seems to have been resolved with the court order. As for the Caretaker FIP benefits that will depend on how the state is reporting that to you if they are.*

she has to have lived with you 6 months, she didn't, no

and no the caretakers money is not income to you