The questions raised are real but can't be answered by giving rules of thumb, I'm afraid. Mathematics is hierarchical by nature and has a long history, with results often building on earlier ones. Peer review of published work varies a lot in thoroughness, but even done conscientiously can't root out all subtle errors. Most of us bring a bit of skepticism to results we haven't understood deeply on our own. Many of us publish results which are not quite right (making later corrections when feasible). It's always risky to quote stuff at random from areas you are not a specialist in, even if you trust the people involved. If you are a specialist, you probably trust some people more than others to get it right; but even so you try to check details. A great many mathematics papers contain at least minor errors, in most cases correctable but not self-correcting. Some real mistakes become famous and spawn important research.
You write: How to decide? Some cases are clear-cut (e.g. most people would accept the classification of finite simple groups), while others are borderline.
Actually, most people I know accept the classification only conditionally, as do some of the real experts in the subject who are still working to codify a complete proof. A result like this, plausible as it looks, depends on a huge amount of published (and some unpublished) work. I think it's still customary to point out in papers when the CFSG is being used to get other results.
If you have to quote results you haven't understood from scratch, you should try to access the MathSciNet database in order to read a review and follow up later citations. Or try to ask an expert. Lacking that, quote at your own risk and without offering too firm an endorsement, e.g., "Theorem X in paper Y states that Z".