Does AngelScript support async compilation of scripts? In other words, can I compile a script on a background thread while my main thread keeps updating UI, graphics etc.?

Yes, there is no problem in running the script compilation in a background thread while the application performs other actions. You can however only compile one script at a time.

Could you clarify? What's meant exactly by other actions? Can code be executed inside an engine while a script is being compiled by the same engine on another thread?

Yes, it is possible to execute scripts with the same engine that is currently compiling another script in a background thread.

If you haven't done so already, make sure you read the recommendations in the manual for multithreaded applications.