If you plan well in advance and have the reservations agents type everything into the PNR, then you should have no problems with the airport agents. Just arrive early and prepare to explain the situation. As far as the return goes, just have the grandmother accompany your child to check-in, and explain from there.
While we're on the topic of minors traveling alone:
I personally prefer AA's policy, allowing solo travel for 15-17 year olds without requiring formal UM service. I know plenty of street-smart 15 year olds who fly all the time and who, frankly, are much more savvy than most Kettles out there when it comes to the intricacies of travel.
But a 13 year old? I'm not sure I would allow one to travel alone without requiring UM service. I'm a high school teacher, and I think there is a vast difference between a 13 year old and a 15 year old--in terms of judgment, decision-making, emotional maturity, etc.
I don't think it's a safety issue. These days, airports are pretty safe places. But that wouldn't stop a child from becoming overwhelmed by the hoards of people if he/she gets momentarily disoriented, or the flight is delayed causing a missed connection, etc. You'd be surprised how quickly young teens can lose emotional control. I've seen it happen for the slightest of things. Now, if you translate that fragility to a real-world environment like an airport, where they have to get to connections, or make decisions about alternatives, I think it could be a recipe for an unpleasant situation for both you and the kid.
Perhaps others feel differently. Of course, nowadays with cell phones in everyone's hands, minors can travel and feel very comfortable having their parents just a push of a button away. But still, I think 13 is too young to be wandering around an airport like DEN by herself.
Others?