The WP Rest API sends the total count(found_posts) property from WP_Query. in a header called X-WP-Total.
FOR POSTS: you can make a call to posts endpoint of the REST API
http://demo.wp-api.org/wp-json/wp/v2/posts
The value for posts count is returned in the header as X-WP-Total. Below is a sample response from the hosted demo
Access-Control-Allow-Headers:Authorization, Content-Type
Access-Control-Expose-Headers:X-WP-Total, X-WP-TotalPages
Allow:GET
Cache-Control:max-age=300, must-revalidate
Connection:keep-alive
Content-Encoding:gzip
Content-Type:application/json; charset=UTF-8
Date:Wed, 28 Dec 2016 12:48:50 GMT
Last-Modified:Wed, 28 Dec 2016 12:48:50 GMT
Link:<https://demo.wp-api.org/wp-json/wp/v2/posts?page=2>; rel="next"
Server:nginx/1.4.6 (Ubuntu)
Transfer-Encoding:chunked
Vary:Cookie
Via:1.1 dfa2cbb51ec90b28f03125592b887c7d.cloudfront.net (CloudFront)
X-Amz-Cf-Id:ri4C3e-AdixwqGv_wYNdGRq9ChsIroy1Waxe2GqkiTqbk4CpiSIQfw==
X-Batcache:MISS
X-Cache:Miss from cloudfront
X-Content-Type-Options:nosniff
X-EC2-Instance-Id:i-198c7e94
X-Powered-By:PHP/7.0.11-1+deb.sury.org~trusty+1
X-Robots-Tag:noindex
X-WP-Total:71
X-WP-TotalPages:8
NOTE:
You can also limit the posts per page you’re fetching to 1 so you’re not getting all your wordpress posts just to get the posts count
http://demo.wp-api.org/wp-json/wp/v2/posts?per_page=1
To Get All Categories
All you have to do is make a GET request to the categories endpoint at:
http://demo.wp-api.org/wp-json/wp/v2/categories
This would return all the categories and also the total count of categories can be found in the X-WP-Total header.