Really nothing to be confused about. It could only matter if the payments from the state were specifically designated for rent. As I understand it, you say they are not designated for anything beyond finding employment (i.e.; employment assistance). So if your daughter is just living with you while seeking employment so she can find her own place eventually, I'd just consider it a cost sharing arrangement where she's just helping with the increased costs of food, utilities and the such while she's living there.
So depending on the legal requirements you daughter may be required to report the income on her own tax return if she's required to file one. (If she got a tax reporting document from whatever entity pays her the money, that's a good indication it's required to be reported on her tax return.) But has far as what she pays you in such a situation, I'd probably consider it as just "helping out" with the expenses. I wouldn't worry about it and not bother reporting anything on SCH E
There are some situations for some types of assistance payment where the recipient has to "prove" their expenses to justify the amount of assistance they receive. Generally one form of proof is a lease agreement. If something like that was in play, then you'd need to report the rental income.
