Check whether set of strings is prefix-free via lexicographic sort?
Yes. The common prefix is maximal with the closest predecessor (if you're treating the empty string as less than any other, which you are).  Edit:  in general if a prefix key exists it will be the immediate predecessor or there will be another key in between which shares the same prefix, and your test will fail on that.  eg ABC, ABCD, ABCE will fail on the first pair even though the last key also contains ABC and is non-consecutive.  The LCP is indeed maximal with the closest predecessor neighbor, but that predecessor neighbor may not be a prefix; I just wanted to clarify that.  
There are situations where the longest common prefix is useful; in particular to augment a suffix array, which is a sorted list of suffixes.  Look up LCP Array. The LCP+1 (roughly) is called the distinguishing prefix; string sorting algos express their complexity in terms of O(n log n + D), where D is the sum of all the distinguishing prefixes of all the keys.  So it's a familiar idea.  
"Prefix free" is the term that I remember.  
You can put them in a trie and look for strings that end on an intermediate node.  It's a variant of sorting.