Any thoughts on whether a capias pro fine hearing pursuant to CCP 43.03 or CCP 45.045 where a person has not paid their fines and costs would be subject to the 6th Amendment right to a public trial?

The imposition of the fine is a final judgment, meaning he already had his trial, right?

There has been a final judgment. But, the capias pro fine process involves the risk of a person's further incarceration. The 6th A guarantees an accused the right to a public trial in all "criminal prosecutions."

For example (although I have not found an answer to this issue), I would assume that one would have the right to a probation revocation hearing be held in open court. Right?

Finally, the capias pro fine process discusses a hearing taking place before the court before incarceration in satisfaction of the outstanding monies may be ordered. Would a "hearing" under this provision by definition be open to the public?

Good question. I'm not aware of any litigation on that. The litigation I've heard about has involved counties that failed to provide an attorney for those convicted of class c offenses who were facing jail time for not paying the fine.

I don't think this is a criminal matter. I think it is most comparable to civil contempt:

Contempt is further classified into either civil or criminal contempt. In determining whether contempt is civil or criminal, it is necessary to examine the purpose behind the contempt order: civil contempt is “remedial and coercive in nature”—the contemnor carries the keys to the jail cell in his or her pocket since the confinement is conditioned on obedience with the court's order, Ex parte Werblud, 536 S.W.2d at 545,6 while criminal contempt is punitive in nature—“the contemnor is being punished for some completed act which affronted the dignity and authority of the court,” id. Thus, the distinction between criminal and civil contempt does not turn on whether the underlying litigation is civil or criminal, but rather on the nature of the court's punishment. Ex parte Chambers, 898 S.W.2d at 266 (Gonzalez, J., dissenting); see generally Ex parte Werblud, 536 S.W.2d at 545–46.
I dunno, seems like a pretty good argument.