I asked AC a few months about what happens to a non-refundable domestic ticket if it isn't used, and I was told you get charged the $145 up front. They then give you the value of the ticket as credit. So it sounds like the second agent was correct. However, I've never tried to do this, and a thread about a week or so ago suggested that they have different rules for each type of ticket. I suggest you call back one or two more times and see if you get the same answer.

Even if you have to pay $145 to IAH, that's still a lot better than losing the ticket.

You definitely will need proof if you try to claim a death.