I don't think so, but again I'm not an expert. Carter's situation would not warrant a sixth year. No chance.
And, you can absolutely take a medical redshirt if you have not taken a regular redshirt. In fact, that's the hole purpose of a medical redshirt. A medical redshirt comes when you've played a small part of the season, then get injured. If you meet qualifications, you get a medical redshirt.
A regular redshirt means you didn't play a single play in a year. Therefore, it would actually be impossible to take a regular redshirt in a season in which you suffer a season-ending injury in game one.
The issue again is this...you cannot take a medical redshirt if you've already used a regular redshirt. I'm almost certain of that. The only exception - as I noted - is if you take a redshirt, then miss two seasons. Then you can ask for a sixth year.