Is there a way to tell whether a paper has been rigorously peer-reviewed and is completely valid? When is it safe to assume that important theorems in a paper you're referencing are valid?  I have a paper in a field in which I am not an expert, so could only check their proofs after months or years of studying other papers which too require their own prerequisites.  How do professionals handle this?  Is there some feature on journal or indexing sites that indicate that a paper is likely valid?
Thanks.
 A: Candidly, this is potentially a serious problem. Journals these days specifically deny that the referee is responsible for certifying "correctness", saying that it is the author's responsibility, etc.
The real point is that unless/until a result is in some way controversial, scandalous, "important", no one cares at all. As in "whatever". 
Needing citations to more prestigious journals is safer in that the results had some other cachet, some claim-to-fame, so were probably doubted/vetted by people more than papers appearing in even the most solid "second" journals. In the latter, or "in life", innocuous things are left alone... and/but if an innocuous thing proves to have some scandalous outcome (your paper?), it will be revisited.
Oop, then no amount of prior passive vetting would be sufficient...?
Or "there are no rules". Being persuasive, solid, is obviously the issue. Published papers are often wrong, and most often that is irrelevant to everything else in the world.
A: You can try searching for the paper(s) in Google Scholar. Each individual result shows how many papers cite it, if it was cited. I am not sure how broad Google's article data is so your mileage may vary.
