How can I get count of downloaded artifacts in specified time period?

You can use the “File Statistics” REST query the get the file statistics, however, you cannot filter by date using this query.

A solution for this can be by writing a user plugin that will implement a beforeDownload execution point and save the download information in a local file or in a DB schema (e.g. add raw with download by, date, user etc…). Once you have the relevant information, you can have another execution point that will take a start date, stop date, and path as parameters and return the relevant information based on the local file/db information that you generated and saved earlier.