No 16 year olds, but I was 17 when I pledged my local sorority. I didn't have to have my parents cosign my financial documents. (In Massachusetts, a 17yo can have her own checking account. Not to mention, a local can set whatever rules they want.
The treasurer probably could have tracked down my parents if I didn't pay, but as a local, our dues were practically pocket change, so I never had a problem meeting my financial obligations.)
We didn't have a house, so no issues there. No risk management issues either, as we didn't have alcohol at our events, and if we had an event such as a mixer where alcohol was available, all underage members (pledges and sisters) were expected to stay away from the booze.
AEPhi does require a parent or guardian to cosign financial documents if a new member is under 18. I have no idea if there are issues with women under 18 living in-house, as we didn't have a house while I was an active.
I do know of a 16yo young man who pledged a fraternity at my school and lived in house, but this was decades ago, and this is hearsay - I didn't know the young man in question.