For example, a scripting language can be an addition to game engine. Or it will allow to write a whole game in the series of scripts. What features are required in such a language for you to use it?

There was a thread about this a month or so back. Suggest you have a search for it, might be useful.

The obvious feature would be that you can give the script as much control over the game as you wish.

I''d use a script to interact with a hardcoded game engine. Eg: The script would be responsible for content, not dictating the gameplay itself. But that''s just me. This would allow the script to operate within predetermined bounds and not, for exmaple, start messing with internal game dynamics