How should I interpret a journal rejection of "not of sufficient interest" or "does not meet journal standards" without mention of any errors? 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?
When a journal rejects a paper because it is not of sufficient interest for them or it does not meet the standards of the journal, it generally means exactly what they said.  They thought the paper wasn't interesting or important enough.

Journals vary enormously in how selective or prestigious they are.  Some journals will accept any paper that seems to be correct, original, and at least somewhat interesting. If a journal like that rejects a paper for this reason, then either it's not a very good paper, or they are being unfair (perhaps accidentally - peer review isn't perfect, so occasionally you just have to try again), or you submitted it to the wrong sort of journal (the line between different subfields can be blurry, but if you choose the wrong side it decreases your chances of acceptance).

Other journals impose extremely high standards and only want to publish papers on exciting breakthroughs.  In that case, there might be nothing wrong with the paper at all, and the only issue is that they have received other submissions they like even better.

And, of course, there's a whole range of journals in between these extremes.  Depending on where the journal is in this range, it will shift from a statement about your paper to a statement about the journal's high standards, and there's no way to be more precise without knowing more about the situation.

The best source of advice is a trusted mentor in your field, but I wouldn't get too worried on the basis of one rejection.  Look over the paper again with fresh eyes, make sure the introduction and conclusions are compelling, choose another journal, and resubmit.  If you run into this problem repeatedly, then something's wrong (either the paper needs work or you need to choose more appropriate journals) and that would be a good time to seek more detailed advice.