There currently are small differences between search api and count, exists, validate query, explain api when it comes to reading query_string parameters. `analyze_wildcard`, `lowercase_expanded_terms` and `lenient` are only read by the search api and ignored by all other mentioned apis. Unified code to fix this and make sure it doesn't happen again. Also shared some code when it comes to printing out the query as part of SearchSourceBuilder conversion to ToXContent.
Extended REST spec to include all the supported params (some that were already supported weren't listed), and added REST tests (also some basic tests for count and search_exists which weren't tested at all).
Closes#11057
Index, type and id were returned as part of the REST explain api response, but not through java api. That info was read out of the request, relying on the fact that the index would get overridden with the concrete one within that same request.
Closes#7201
