Achilles is for database characterization, so it inspects the database as a whole (ie: no sub-groups like men, age > 50). Those dropdowns are static, they are based on the domains of the CDM.
The reports you find in Achilles UI are pre-canned analysis reports (such as the gender breakdown, or the condition drilldowns) which simply read from a static .json file and render the reports as HTML. There's nothing to query there, tho it's technically possible to do some client-side filtering, but I don't think it would get the effect you want.
If you are interested in exploring a cohort's baseline characteristics, I suggest you define your cohort (for your example you could create a cohort of 'visit occurrence, gender is male, age is > 50, limit to earliest per person), and then use the 'generate with features' function in Atlas, and then you can see things like conditions, drug exposures, age groups (which would be, in this case, only the 50-55 group).
But if you are looking to work with the functionality directly out of Achilles, you should review the queries that are executed (they are grouped by an analysis_id range, example: 1-12 is general counts, 101-120 is some demographic statistics, etc). The raw queries are here: https://github.com/OHDSI/Achilles/blob/master/inst/sql/sql_server/Achilles_v5.sql. The 'raw results' that get inserted into achilles_results should be granular enough that if you wanted to create your own reports, you can query those results tables directly, looking for the specific analysis_id that contains the statistic you require, and then do the appropriate aggregations. Of course, this requires SQL knowledge. If you wanted to make it easier for your users to not know SQL, you could create your own php-based web application that queries those tables on the user's behalf.
But, I think you should try out the 'feature generation' functionality; Here's an example on the public site: http://www.ohdsi.org/web/atlas/#/cohortdefinition/3134. Just navigate to the 'generation' tab, and then you can click 'view reports'. The baseline characteristics is under the 'Chort Features' tab at the bottom.