Some of the data you are looking for is available at /about.json. It will give you the total number of posts and topics, and also data for the last 7 days and the last 30 days.
To get the post count for a specific time period, you can make an API call to any of the admin reports that are displayed on your site’s dashboard. For the posts report, you can make a query to /admin/reports/posts.json. The start_date and end_date can be set as query parameters.