When I use Search and implement my own autocomplete functionality, I use queries for each set of autocomplete results. Can I use the autocomplete feature of Acquia Search powered by SearchStax in a similar manner? Does each response count as a query?

Yes, each response counts as a query. However, there is a different endpoint and index for autocomplete, so it does not contribute to the count of search queries. There are two indexes, one for data that the Drupal application connects to, and another for autocomplete. If you use the autocomplete index, the response does not count as a search query. However, if you continue to use the main index, the response counts as a search query.