Hi Petr! Welcome. The first link you shared appears to have a table of contents linking to headings within the page. This is something you can achieve with DiscoTOC - automatic table of contents.
The second screenshot is of search, which Discourse does very wel! Try it out here on meta, by selecting the 
in top right corner.
I’m not quite sure what you mean by “link it in with an in-app widget”. Can you explain in some more detail?