Should there be a theorem or proposition that uses lemma?
But is it enough if I use some lemma in an example, or should there be a theorem or proposition that uses that particular lemma?


I don't think it really matters.  An example would be fine for most readers, while a few really formal people might disapprove.  However, even someone who disapproves wouldn't consider this seriously problematic.  I.e., they wouldn't think less of you or negatively evaluate your paper because of this.  The worst case scenario is that someone complains about it in a referee report, and even then you wouldn't have to follow their advice.