EDIT: Answers in brackets
What would the cousin's position be? Would she be a companion? Is that acceptable when the companion is younger? (Yes) Afaict the two reasons a companion was usually older was for education and as a chaperone, both of which are unnecessary. Would it be ok if she was just a friend or would it be expected than an actual companion was hired? What if the mother dies?
If they do unexpectedly go out into society, would the older woman be an acceptable chaperone(No), or would they need someone married or older still? (Yes) Is there a rough age when spinsters were considered old enough to be chaperones? (Well over 28?)
EDIT: So it's ok for her to be a companion, but they'd need a married female friend/relative as a chaperone if they ever went to a ball or something. Yes?
I haven't been able to find any information not contained in the wikipedia page on Lady's Companions, searching for "Lady's companion" and "Lady's companion regency" gets me redundant information and lots of creepy dating ads :/ The tags here didn't seem to have anything either.