get all sections separately with wikimedia api

What you ask is called parsing, because it requires interpretation of the wikitext source to split the page by sections etc. So the solution is given in https://www.mediawiki.org/wiki/API:Parsing_wikitext

1) Get the list of sections: https://www.mediawiki.org/w/api.php?action=parse&page=API:Parsing_wikitext&prop=sections

2) Ask the parsed wikitext of that section: https://www.mediawiki.org/w/api.php?action=parse&page=API:Parsing_wikitext&section=1&prop=text

I realize this question was asked four years ago, so possibly the following was not available then:

You can use the REST API described here: https://www.mediawiki.org/wiki/REST_API

The REST endpoints are described/documented here: https://en.wikipedia.org/api/rest_v1/#/

The mobile-sections endpoint (intended for consuming info for a mobile device) gives you a nice breakdown with headings, which sounds like what you are asking for.

Alternatively, the metadata endpoint returns a toc (table of contents) section which contains the same breakdown of headings.

Here is an example URL, fetching the mobile sections for the "Egyptian pyramids" page: https://en.wikipedia.org/api/rest_v1/page/mobile-sections/Egyptian_pyramids

The advantage is that the response is in JSON format (which is what you were asking for).