1. Yes, in that case ZFC would be inconsistent.
2. Yes, it is theoretically possible. As you yourself mention, if ZFC happens to be consistent, then it cannot be proven to be consistent from within ZFC itself. However, if it is inconsistent, then there will be some statement, such that both the statement itself and its negation are true, and it is certainly possible that you can find proofs of both the statement and its negation.
3. If ZFC is consistent (without us knowing), then we will strictly speaking never know for sure, whether it is consistent or not. However, in an inconsistent system, you can prove that _all_ statements are true statements. My thoughts are that if ZFC was inconsistent after all, then since all statements are true statements, we should by now have come across two proven inconsistent statements, i.e. two inconsistent statements that have both been proven to be true.