Is it possible even for a SharePoint noob like me to make up a page with many links (which only return a simple JSON result like true or false) and show the result of the clicked link on the same page next to the link? I think this is a rather common use case and was hoping there is something out of the box but don't know where to look.
I assume you are trying to call Rest API with some links and retrieve the results to a filed in the page to display. Per my knowledge there are no out of the box web parts doing such work. The links web part holds the links and open the new page in a new tab.
If you are using SharePoint On-Premise, you may need to use a content editor/script editor web part to hold scripts or simply create a custom web part for your goal. Here is an example which you can have a check:https://social.technet.microsoft.com/Forums/ie/en-US/d9a06d93-93d2-4340-a491-c7d0d5d4fdf3/how-to-dynamically-change-link-in-page-viewer-web-part-sharepoint-2010?forum=sharepointgeneralprevious
For SharePoint Online, you may need to use use SPFx to do so.