using imagelist in script form

Is it possible to register an imagelist instance for scripting and use it inside the script? 

For instance, I want a TButton use that imagelist by setting its Images and ImageIndex property. 

Yes, it's possible, and you don't need to register, as it's registered by default. If you launch the scripter IDE (for example, the ScripterProIDE demo in "demos\ide" folder), you will see TImageList is available in the component palette.

So I can drop a TImageList instance onto a script form - that's fine.

Is there a possibility to use a TImageList instance available inside the program (a Delphi object, not in the script)? 

I already found that I can add that instance to the script and then assign it in the FormCreate event of the script. Is there a possibility to make this instance available inside the Scripter-IDE?

No, the components in the script form are just like in Delphi: they are "design-time" and whenever you create an instance of the form, instances of the components dropped in it are created. You can use existing ImageList (and any object) from script, but then only from code, at script "runtime".