? Does anyone know where (if it exists) the statute is located that basically says a criminal defense attorney who is found to be ineffective on a capital murder case is barred from representing defendants in the future on capital cases?

Second, please share your thoughts on whether or not a civil action against the ineffective attorney(s) for fees paid by the county would be viable or advisable.

Look at Article 26.052(d)(2).

I doubt that suing the defense attorney is a good idea. Might be politically popular, at least in the short term, but it raises many problems and doesn't offer much in return. It should be pointed out that courts often find "prosecutorial misconduct" and other bad behaviors (like 'Batson' violations) on the prosection side. And there are always people quick to blame "ineffectiveness of prosecutor" every time there is a 'not guilty' or a dismissed charge.
Inviting the kind of civil war that civil suits would entail is unlikely to help anyone in the long run. There is an exception for outright corruption that might also qualify as a crime (i.e. rises to the level of fraud, etc.).

If you think the circumstances warrant it, you can forward to the Bar for review/discipline. Bar sanctions are probably far worse than having to pay back any indigent compensation (which is not that great). Many jurisdictions already have problems getting qualified attorneys to take indigent capital cases, and the "prevailing professional norms" of capital defense are much in dispute. In addition, on pragmatic grounds, if you did file such a suit you oould expect it to be fought to the ends of the earth.

Thanks for the reference and comments - well put.