1. No
2. No. You can potentially upgrade 1 segment (TATL) if there is at least a single AA marketed segment. For example, if you are on BA 138 connecting to BA 1504 (operated by AA), you cannot. If you are on BA 138 connecting to AA 51, you can.
3. EF does not work for connecting flights. The C on each segment is correct. If it is C0 on LHR-DFW, then that is what it is.
Also, C space on BA operated flights is meaningless. Exclude codeshares for accuracy.