Some of these are also on your FBS teams list. Also, your naming system is really inconsistent. For example, University of Tennessee at Martin is "UT-Martin", but "UT-Chattanooga" doesn't work. I know this is a lot of work, just trying to give you some feedback.
I think you're confused on the first point. There are no teams that are in both FBS and FCS, and no teams (that I can see) that are in both of my map quizzes.
On the second point, the names shown are Wikipedia's consensus common names for the schools. UT Martin brands itself that, and is called that in the majority of FCS media coverage, and same for Chattanooga. I can add UT Chattanooga as a type-in, but it is not the most common way that the school is named by reliable sources related to football.
Tarleton State, Dixie State, North Alabama, and Merrimack are all recent additions to FCS football, and I believe all are planning on playing the 2020 spring season.
On the second point, the names shown are Wikipedia's consensus common names for the schools. UT Martin brands itself that, and is called that in the majority of FCS media coverage, and same for Chattanooga. I can add UT Chattanooga as a type-in, but it is not the most common way that the school is named by reliable sources related to football.