We already provide analytics about the searches performed on your indices.
However, we focus on user-facing search use-cases. We don't recommend using Algolia for processing your logs for example. See this quora answer for an Elasticsearch / Algolia comparison:
http://www.quora.com/How-does-Elasticsearch-relate-and-or-co...
Yes, we already existed before operating the new HN search. We did it as a way to participate to the community and of course it's good for our visibility. The hnsearch.com domain is the property of Octopart, not ours.
We usually recommend to perform one query (one API call) per keystroke starting from the first one. The actual number of calls depends a lot on the use-case. Our ranking takes into account both relevance and popularity to suggest the best result first which greatly reduces the number of letters you need to type.
In use-cases where there is a very strong popularity indicator, like the number of followers for TV shows, we usually get the correct result at the first keystroke (b -> breaking bad, d -> dexter). At the other extreme, you may need to type several words.
Great feedback. Your UI comments in particular are insightful. It's impossible to get something that pleases everyone, but we'll try to get it closer to current HN design in the next iteration.