Redundancy is one big source of self-healing. A result with three different proofs is rather unlikely to be wrong. Also, people try to apply fresh results; wrong results often lead to contradictions when applied, alerting mathematicians to their wrongness. Same for proofs: Mistakes in proofs are often spotted when someone tries to adapt the proof to other questions.

This is tricky. These days, using Google Scholar's "cited by" feature and various other backlink aggregators, you can get a list of papers/book that reference a given paper. Thus, if you find an error in the literature, you can track down where the "corruption" has spread. But getting corrections published is very difficult. Ted Hill and Nikolai Mnev are known for having struggled through the whole process of correcting someone else's false claims, but lots of people end up staying silent or (these days) just posting what they know somewhere on a forum like MathOverflow when someone stumbles upon the same problem. Then there are situations where no specific error can be pinpointed, but important material is simply imprecise and unreadable; fields often linger in such a limbo until someone does the thankless job of building the foundations underneath them. Katrin Wehrheim is one example of this.

This question of mine got 41 votes, so yes, this is a fairly well-acknowledged problem.

Ask your advisor and others. You definitely want to understand all proofs in undergraduate and lower-level graduate classes; they aren't particularly likely to be wrong, but you'll use the ideas anyway. As for advanced theory you rely upon, it depends.