Q. Does having more unemployment than $4300/yr exclude her from being claimed even if she is a full-time student?
A. Yes and no. It excludes the father from claiming her. It does not exclude you from claiming her. There are two types of dependents, "Qualifying Children"(QC) and standard ("Qualifying Relative" [QR] 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, student status, a relationship test and residence test. You meet the QC rules because the child lives with you. The father does not. The father does not meet the QR rules because of the $4300 income test (and probably a couple of other tests).
Q. Can her father claim her because she is a full-time student?
A. No. The full time student rule only applies if the student lives with the parent. You can still claim her under the Qualifying Child rules, but the father can't. An "apartment in the college town", generally means a temporary absence from the parent's home. If she was living with you, before she went to school, she is still considered as living with you. Although, "she's not even coming home for breaks" does make that iffy. She cannot be considered as living with her father.
Unless you live in MS, she is considered emancipated, by age alone, for purposes of the special tax rules for divorced or separated parents.
