In regards to whether or not Bishops approve of apparitions, is their judgment final and are they incapable of error?
Approving or disapproving apparitions is not an infallible decision because even if an apparition is “approved”, meaning worthy of belief, the faithful are not obliged to believe them.
The normal process in a nutshell is that should someone claim to have seen an apparition the local Bishops would undertake an investigation. If they decide something supernatural has occurred it would then be moved on to the CDF (Congregation for the Doctrine of the Faith) in Rome who in turn would investigate and come to a decision to approve or not. They do not have to rubber stamp the local decision.
On the other hand if the local Bishops decide there is nothing supernatural taking place or there is no evidence of any supernatural happenings then it ends there. It does not normally go to Rome.
An exception is Medjugorie. Three local Bishops Commissions at different times have all declared there is no evidence of any supernatural happenings. They have gone further to say the alleged seers are frauds. However, because of the many people who still believe and the controversy caused this one has gone to the CDF. Since then local priests who have been perpetuating the apparitions as true have had their faculties suspended and they cannot say Mass or hear Confession. The final step will be a decision by the CDF which is almost certain to be a disapproval.