To stray from mathematical logic to how other mathematicians might think about proofs...
I think many mathematicians go with Carl's "convince ourselves of their consistency by methods that are not completely formal." Many mathematicians use set theory simply as a language--probably similar sorts of mathematicians as do not concern themselves with categories too much, of which type there are still many. Mathematicians with a physics bent often enjoy "informal" arguments based on physical intuition, a mechanical construction, or the nonrigorous arguments of Archimedes or Appolonius or Cavalieri using a primitive version of infinitesimals to compute volumes, etc. The insight gained from less formal arguments, while less definitive, perhaps, probably outweighs the worries about set-theoretic and proof-theoretic issues for many mathematicians. (A graph theorist, for example, could be perfectly happy proving results for classes of graphs and graph properties for their whole career, knowing that the results are true for graphs the way one usually pictures them, without worrying about the consistency of ZFC).
Since your question is part mathematical logic and part psychology (whether people "worry"), may I suggest some of the literature on pedagogy for higher mathematics? Many people have thought a lot about how to treat the concept of proof and other issues in courses for various sorts of students in order to maximize the understanding and value gained. See, for example, David Henderson's work on "educational mathematics" at Cornell.
