Do Extract APIs execute Javascript?

Yes. Diffbot Extract APIs execute page-level Javascript at render-time, and for the most part will be able to access Ajax-delivered content.

Note that while creating a custom API you will sometimes not be able to preview Javascript-delivered content, but it will be accessible when making actual API calls.

Crawl does not execute Javascript when harvesting/collecting links by default, but it can be modified to do so while using the Analyze API. See How to find and access JavaScript-generated links while crawling for details on enabling this functionality.