Sorry, I think I misread your original description.
I guess I missed the "paste" part. If you want to copy/paste something from an iPad application to an iPad browser (since you cannot copy on one device and paste on another without something like cloud clipboards), you cannot do that with extensions because iPad browsers do not support Chrome extensions at all (as mentioned).
Also, extensions cannot take a screenshot of other applications (you can ask the user to share the entire desktop/specific window and take a snapshot of that video, though, but websites can also do that), nor can they control other applications (unless the application exposes an HTTP based API or a protocol handler, but then websites can also control them).