This is a delicate situation and I don't want to sound rude or uncaring so please don't take it that way.
In your phone conversation with DL to make your cousin more comfortable on the way home, you asked for an aisle or exit row seat or a first class seat. 2 of the 3 very reasonable requests. What was the response by DL when you asked for an aisle or exit? Just that they couldnt' do it? I can kind of understand why they declined an exit row seat but if there was an open aisle seat, I see no reason not to move her to one.
Also, when you asked for the supervisor's supervisor (which would actually be a manager) and you were told there is no one higher than her/him to speak with, why would you flippantly ask if he/she owned DL?? That comment will put the agent on the defensive and will do little to help the situation. Manager's (when they are there) generally don't take phone calls or deal with the public. I say generally because it does happen but for a very minor request like a seat change or upgrade which is pretty cut and dried, there would be no reason for a manager to become involved. Either the seat can be moved or it can't. A manager will have no more of a way to create a seat than an agent or supervisor.
The first class seat is the issue I really wanted to address. Delta said she could upgrade if you/she paid extra or used more miles, which it sounded like you ended up doing. But then you go on to say you were ok with that and later mention you weren't asking for a free ride. If you received the upgrade after using miles and you didn't want it for free, what is the complaint?? DL assisted you with what was available and it seems like a pretty normal transaction.
After all is said and done, I hope you and your cousin are doing well are both on your way back to good health.
