What are some standard ways of computing the distance between individual search queries?
The cosine similarity metric does a good (if not perfect) job of controlling for the document length, so comparing the similarity of 2 documents or 2 queries using the cosine metric and tf idf weights for the words should work well in either case. I would also recommend doing LSA first on tf idf weights, and then computing the cosine distance\similarities.

If you are trying to build a search engine, I would recommend using a free open source search engine like solr or elastic search, or just the raw lucene libraries, as they do most of the work for you, and have good built in methods for handling the query to document similarity problem.