I am an attorney, but this is not legal advice. If you need legal advice, pay my retainer, cheapskate!
>The only special case is a verdict of innocent in a criminal case.
There is no such verdict (except, I believe, in Scotland). The verdict is "not guilty," which includes "we're pretty sure he did it, but not beyond a reasonable doubt."
The legal decisions of a judge are reviewed "de novo" by the appellate court--no (zero, zilch, nada) weight is given to the judge's own decision, save for its ability to persuade the higher court by argument.
Findings of fact, though, whether by judge or jury, are appealable, but the standard is roughly whether any reasonable person could have reached that conclusion from the evidence given; the appellate court does not substitute its own judgment of the facts.
(There are also a range of areas, such as remedies and sanctions, which are a mix of law and fact, in which a judge will only be overruled for "abuse of discretion.")
hawk, esq