How should I interpret a journal rejection of "not of sufficient interest" or "does not meet journal standards"?
This is what happened to me particularly. Papers were rejected for those reasons and the journal never told me that they had found any error or that anything was wrong or inconclusive.
When a paper is rejected, do reviewers let you know if they found any error or they will never tell you even if they found one?
The short answer is that if a reviewer found an error, the reviewer will generally tell you. If you get a rejection without any further comments, the likely reason is that the reviewer read the outline and main results, and concluded that it wasn't necessary to go through the paper carefully to decide to reject it, probably because the results weren't significant enough for that journal. (Some journals specifically request that reviewers do a quick read of the article within a couple weeks of receipt, to see if it has any hope of being published; it often takes reviewers months to do a full read through, and if it has no chance, it's kinder to the author to give a quick rejection so the author can promptly resubmit to a journal which might publish it.)
However sometimes the reviewer has carefully read through the paper before recommending rejecting it, and in that case the reviewer usually (at least in my experience) includes a list of suggestions or comments (including pointing out any errors the reviewer found).